Interval notation is a way to represent an interval of real numbers on the number line. Set notation is a way to represent a set of elements.
What is interval and set notation?Interval notation is a way to represent an interval of real numbers on the number line.
The notation uses parentheses, brackets, and infinity symbols to indicate whether the endpoints of the interval are included or excluded from the set of numbers.
For example, [3, 8) represents the interval of real numbers from 3 (included) to 8 (excluded), while (-∞, 4) represents the interval of real numbers less than 4 (excluding 4), and extending to negative infinity.
Set notation is a way to represent a set of elements. It uses curly braces to enclose the elements of the set and can include various symbols to indicate properties of the set.
For example, {2, 3, 5, 7, 11} represents the set of prime numbers less than 12, while {x | x is an even number} represents the set of even numbers.
The vertical bar | is used to separate the variable (x in this case) from the condition that must be met for elements to be included in the set (x is an even number).
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Can you help me with this?
16. The equatiοn οf the line in slοpe-intercept fοrm that passes thrοugh the pοint (-6, 5) and is parallel tο x + 2y = 14 is y = (-1/2)x + 2.
What is equatiοn οf line?The equatiοn οf a straight line is y = mx + c, y = m x + c m is the gradient and c is the height at which the line crοsses the y -axis, alsο knοwn as the y -intercept.
16. Tο write the equatiοn οf a line in slοpe-intercept fοrm, we need tο find the slοpe and the y-intercept οf the line.
Tο find the slοpe οf the line, we can rewrite the equatiοn x + 2y = 14 in slοpe-intercept fοrm y = mx + b by sοlving fοr y:
x + 2y = 14
2y = -x + 14
y = (-1/2)x + 7
The slοpe οf the line is -1/2.
Since the line we want tο find is parallel tο this line, it will have the same slοpe οf -1/2.
Nοw we can use the pοint-slοpe fοrm οf the equatiοn οf a line tο find the equatiοn οf the line that passes thrοugh the pοint (-6, 5) with a slοpe οf -1/2:
y - y1 = m(x - x1)
where (x1, y1) is the pοint (-6, 5), and m is the slοpe, -1/2.
y - 5 = (-1/2)(x - (-6))
y - 5 = (-1/2)x - 3
y = (-1/2)x + 2
17. The equation perpendicular to y = -(2/3)x + 4, passing through (-4, 6)
perpendicular equations slope would be negative reciprocal to the current line.
The slope in y = -(2/3)x + 4, is m = -(2/3),
The negative reciprocal of -(2/3) is 3/2
Now, applying the x and y values in pοint-slοpe fοrm
y - 6 = 3/2(x - (-4))
y = 3/2(x+4) + 6
y = (3/2)x + 6 + 6
y = (3/2)x + 12
18. Since the line we want tο find is parallel tο this line, it will have the same slοpe.
Lets find the slope using slope formula
[tex]\rm m = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]
[tex]\rm m = \dfrac{0 - (-1)}{2 - (-1)}[/tex]
[tex]\rm m = \dfrac{1}{3}[/tex]
Now, using the point slope form
y - 1 = 1/3(x - 3)
y = 1/3(x - 3) + 1
y = (1/3)x - 1 + 1
y = (1/3)x
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4. Circle the reason for each of the following manipulations used to simplify the product (8x²)(3x²).
(8.3).(x²-x²)
8x²-3x²
commutative or associative
8.3.x²-x²
commutative or associative
24x²
commutative or exponent property
8.3.x².x² = (8.3).(x².x²) - commutative and associative properties of multiplication.
What is commutative law?Commutative laws deal with arithmetic operations addition and multiplication. This means that changing the order or position when adding or multiplying two numbers does not change the final result. For example, 4 + 5 is 9 and 5 + 4 is also 9. The order in which the two numbers are added does not affect the sum. The same concept applies to multiplication. Commutativity does not apply to subtraction and division, because changing the order of the numbers yields a completely different final result.
(8x²)(3x²) can be simplified as follows:
(8x²)(3x²) = 8.3.x².x² = (8.3).(x².x²)
= [tex]24x^4[/tex]
The reason for each of the manipulations is as follows:
8.3.x².x² = (8.3).(x².x²) - commutative and associative properties of multiplication.
(8.3).(x².x²) = [tex]24x^4[/tex] - exponent property of multiplication.
Therefore, the final answer is [tex]24x^4[/tex].
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You and your siblings decided to make 10 pies for a bake sale. There were 8 slices in each apple pie and 10 slices in each shoo-fly pie. At the sale, there were 84 slices available. How many of each pie were made?
Answer: 6 apple pies and 4 shoo-fly pies were made.
Step-by-step explanation:
Let's assume the number of apple pies made is A, and the number of shoo-fly pies made is S.
We know that there were 8 slices in each apple pie, so the total number of apple pie slices would be 8A. Similarly, the total number of shoo-fly pie slices would be 10S.
We also know that the total number of slices available was 84. So we can write an equation:
8A + 10S = 84
We have two unknowns and only one equation, so we need another equation to solve for A and S. We know that the siblings made a total of 10 pies. So we can write another equation:
A + S = 10
Now we have two equations and two unknowns, so we can solve for A and S. We can use substitution to eliminate one variable:
A + (10 - A) = 10
10 = 2A + 10
2A = 0
A = 0
This is obviously not the solution we're looking for, so there must be an error in our calculations. Let's check our first equation:
8A + 10S = 84
If A = 0, then we have:
10S = 84
S = 8.4
This is also not a valid solution since we can't make 8.4 shoo-fly pies. The mistake we made was assuming that both A and S were whole numbers. We can fix this by using another equation:
A + S = 10
We know that A and S are both whole numbers and that A + S = 10. The only pairs of whole numbers that add up to 10 are (1, 9), (2, 8), (3, 7), (4, 6), and (5, 5).
Let's try each pair and see which one gives us a valid solution:
(1, 9): 8(1) + 10(9) = 98 (not 84)
(2, 8): 8(2) + 10(8) = 96 (not 84)
(3, 7): 8(3) + 10(7) = 94 (not 84)
(4, 6): 8(4) + 10(6) = 92 (not 84)
(5, 5): 8(5) + 10(5) = 90 (not 84)
None of the pairs work, which means there is no valid solution that uses whole numbers for A and S.
However, we can use decimals to get a solution that's close to the desired number of slices. Let's try (4.2, 5.8):
8(4.2) + 10(5.8) = 84.4 (close to 84)
This means that the siblings made 4.2 apple pies and 5.8 shoo-fly pies. Since we can't make a fraction of a pie, we'll round up the number of apple pies and round down the number of shoo-fly pies:
4 apple pies and 5 shoo-fly pies would give us a total of 8(4) + 10(5) = 84 slices, which is the desired number.
In a lab experiment, a population of 400 bacteria is able to triple every hour. Which equation matches the number of bacteria in the population after 3 hours?
Answer:
400(3)^3
Step-by-step explanation:
It tripled for 3 hours which is 3^3 and there's 400 bacteria
Question 3
The Orono Middle School Unified Club earned $175 at a car wash. If this amount is 25% of the cost of new set of uniforms for their next basketball tournament, what is the total cost of the new set of uniforms?
$700. If the costs (C) of the outfits is $175, then we can create an equation that looks like this.
C x .25 = $175
To find C, taking the derivative of the equation by.25, and you get...
C = $700
By multiplying the value by the entire value and multiply that number by 100, the percentage may be calculated.
Sample percentages include:
10% is equal to 10/100, or 1/10 of the total.
20% is equal to 20/100, or 1/5 of the total.
30% is equal to 30/100, or 3/10 of the total.
40% is equal to 40/100, or 2/5 of the total.
50% is equal to 50/100, or half of the number.
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Jamal sold hotdogs at a recent basketball game. Each hotdog sold for $3.50. In total, Jam
sold $98 worth of hot dogs. Let n be the number of hotdogs that Jamal sold.
Set up an equation that models the information given in this problem.
Answer:The concession stand sold
46
hot dogs and
32
hamburgers.
Explanation:
The first thing to do in algebraic problems is assign variables to things we don't know, so let's start there:
We don't know how many hot dogs the concession stand sold, so we will call that number
d
.
We don't know how many hamburgers the concession stand sold, so we will call that number
h
.
Now we translate the statements into algebraic equations:
The number of hot dogs and hamburgers that were sold is
78
, so
d
+
h
=
78
.
If each hot dog is sold for
1.25
, then the total revenue from hot dogs is given by
1.25
d
. In the same way, the total revenue from hamburgers is
1.50
h
. The total revenue from both hot dogs and hamburgers should be the sum of these, and since we are told the total revenue is
105.50
, we can say
1.25
d
+
1.5
h
=
105.5
.
We now have a system of two linear equations:
d
+
h
=
78
1.25
d
+
1.5
h
=
105.5
We can solve it using several methods, though I'm going to go with substitution. Use the first equation to solve for
d
:
d
+
h
=
78
→
d
=
78
−
h
Now plug this in for
d
in the second equation:
1.25
d
+
1.5
h
=
105.5
→
1.25
(
78
−
h
)
+
1.5
h
=
105.5
Solving for
h
, we have:
97.5
−
1.25
h
+
1.5
h
=
105.5
0.25
h
=
8
h
=
8
.25
→
h
=
32
Since
h
+
d
=
78
,
32
+
d
=
78
→
d
=
46
Step-by-step explanation:
Observation what is going on regarding determinant of product of two matrices. al.) Make a conjecture about the relation between det(AB), and det(BA). Type your answer after %. a2.) Make a conjecture about the relation between det(AB), det(B), and det(A). Type your answer after %. a3.) Make a conjecture about the relation between det(A™), and det(A). Type your answer after %.
The determinant of two matrices for the question a1 is it has not been demonstrated, a2 the multiplied matrices can affect the determinant of the product, and a3 A matrix's own determinant will be the same as the determinant of its transpose.
a1.) Conjecture about the relation between det(AB), and det(BA):
The determinant of the product of two matrices is not necessarily equal.
If two matrices A and B are multiplied together to produce the product AB, it is not necessary that the determinant of AB is equal to the determinant of BA.
This is a conjecture that has not yet been demonstrated in every case.%
a2.) Conjecture about the relation between det(AB), det(B), and det(A):
The following conjecture could be made about the relation between the determinants det(AB), det(B), and det(A):
det(AB) = det(BA) det(AB) = det(A)det(B)det(BA) = det(A)det(B)
These conjectures are not true in general.
It is because the order in which matrices are multiplied can affect the determinant of the product.%
a3.) Conjecture about the relation between det(A™), and det(A):
This conjecture about the relation between the determinants det(A™) and det(A) can be made:
det(A™) = det(A)
The transpose of a matrix does not alter the determinant, as long as the matrix is square.
The determinant of a matrix will remain the same if the rows and columns are exchanged.
Therefore, the determinant of the transpose of a matrix will be equal to the determinant of the matrix itself.
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Martin makes b bags of snack mix. Each bag contains 1.5 lb of nuts and 0.75 lb of dried fruit. What does the expression 1.5b + 0.75b represent?
Pls explain why
1.5b + 0.75b represents the total amount of nuts and dried fruit used for b bags of snack mix.
b is the number of bags of snack mix, so the expression 1.5b means 1.5 multiplied by b, which is the total amount of nuts used for b bags of snack mix. Similarly, 0.75b is 0.75 multiplied by b, which is the total amount of dried fruit used for b bags of snack mix.
Adding these together gives us 1.5b + 0.75b, which is the total amount of nuts and dried fruit used for b bags of snack mix.
In order to make the same amount of money, they would have to each sell ______ bicycles. They would both make $______.
In order to make the same amount of money, they would have to each sell 5 bicycles. They would both make $500
How many bicycle would they sell to make the same amount of money?To find the number of bicycles they would need to sell to make the same amount of money,
We can set Jim's and Tom's weekly earnings equal to each other and solve for the number of bicycles:
250 + 50x = 400 + 20x
30x = 150
x = 5
So they would need to sell 5 bicycles to make the same amount of money.
How much would they make for selling that amountTo find out how much money they would make for selling 5 bicycles, we can substitute x = 5 into either equation.
Let's use Jim's equation:
250 + 50(5) = 500
So they would make $500 for selling 5 bicycles.
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a child who weighs 18kg is to receive motrin (ibuprofen) 8mg/kg by mouth every 4 hours as needed for pain. the label reads 100mg/5mls. how many milliliters will you administer?
According to the information on what the child should receive ibuprofen, it should be administered to the child 1.6 mL.
What is Ibuprofen?Ibuprofen is a nonsteroidal anti-inflammatory drug (NSAID). It works by inhibiting the body's synthesis of prostaglandins. This aids in the reduction of swelling, pain, and fever. It can also be used to relieve mild to moderate pain caused by menstruation, arthritis, or toothache.
To calculate the amount of milliliters to be administered, we need to use the following formula:
Amount of Motrin = Weight of child (kg) × Dose of Motrin (mg/kg) ÷ Concentration of Motrin (mg/ml)
Where
Weight of child = 18kgDose of Motrin = 8mg/kgConcentration of Motrin = 100mg/5mlsSubstitute the given values in the above formula.
Amount of Motrin = 18kg × 8mg/kg ÷ 100mg/5mls= 144mg ÷ 20mg/mls= 7.2mls
Therefore, the amount of Motrin to be administered is 7.2mls.
However, the dosage amount administered is not in 5ml increments. Therefore, we need to round it to one decimal place. Thus, we'll have: Amount of Motrin = 7.2 ml (rounded to one decimal place) = 1.6 ml Answer: 1.6 ml
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.......???????????????
Answer:
Step-by-step explanation:
[tex]x^2-5=-7x-1[/tex]
[tex]x^2+7x-5=-1[/tex] (subtracted 7x from both sides of the equation)
[tex]x^2+7x-4=0[/tex] (+1 both sides)
Use quadratic formula to solve for x:
[tex]x=\frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex] where [tex]a=1,b=7,c=-4[/tex]
[tex]=\frac{-7 \pm \sqrt{7^2 - 4\times1\times(-4)} }{2\times 1}[/tex]
[tex]=\frac{-7 \pm \sqrt{49 +16} }{2}[/tex]
[tex]=\frac{-7 \pm \sqrt{65} }{2}[/tex]
[tex]x=\frac{-7 +\sqrt{65} }{2},\frac{-7 - \sqrt{65} }{2}[/tex]
[tex]x=0.53,-7.53[/tex]
Guidance Missile System A missile guidance system has seven fail-safe components. The probability of each failing is 0.2. Assume the variable is binomial. Find the following probabilities. Do not round intermediate values. Round the final answer to three decimal places, Part: 0 / 4 Part 1 of 4 (a) Exactly two will fail. Plexactly two will fail) = Part: 1/4 Part 2 of 4 (b) More than two will fail. P(more than two will fail) = Part: 214 Part: 2/4 Part 3 of 4 (c) All will fail. P(all will fail) = Part: 3/4 Part 4 of 4 (d) Compare the answers for parts a, b, and c, and explain why these results are reasonable. Since the probability of each event becomes less likely, the probabilities become (Choose one smaller larger Х 5
The probability of all will fail is the lowest.
The given problem states that a missile guidance system has seven fail-safe components, and the probability of each failing is 0.2. The given variable is binomial. We need to find the following probabilities:
(a) Exactly two will fail.
(b) More than two will fail.
(c) All will fail.
(d) Compare the answers for parts a, b, and c, and explain why these results are reasonable.
(a) Exactly two will fail.
The probability of exactly two will fail is given by;
P(exactly two will fail) = (7C2) × (0.2)2 × (0.8)5
= 21 × 0.04 × 0.32768
= 0.2713
Therefore, the probability of exactly two will fail is 0.2713.
(b) More than two will fail.
The probability of more than two will fail is given by;
P(more than two will fail) = P(X > 2)
= 1 - P(X ≤ 2)
= 1 - (P(X = 0) + P(X = 1) + P(X = 2))
= 1 - [(7C0) × (0.2)0 × (0.8)7 + (7C1) × (0.2)1 × (0.8)6 + (7C2) × (0.2)2 × (0.8)5]
= 1 - (0.8)7 × [1 + 7 × 0.2 + 21 × (0.2)2]
= 1 - 0.2097152 × 3.848
= 0.1967
Therefore, the probability of more than two will fail is 0.1967.
(c) All will fail.
The probability of all will fail is given by;
P(all will fail) = P(X = 7) = (7C7) × (0.2)7 × (0.8)0
= 0.00002
Therefore, the probability of all will fail is 0.00002.
(d) Compare the answers for parts a, b, and c, and explain why these results are reasonable.
The probability of exactly two will fail is the highest probability, followed by the probability of more than two will fail. And, the probability of all will fail is the lowest probability. These results are reasonable since the more the number of components that fail, the less likely it is to happen. Therefore, it is reasonable that the probability of exactly two will fail is higher than the probability of more than two will fail, and the probability of all will fail is the lowest.
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Bernard's rectangular bedroom is 12 feet by 16 feet. What is the diagonal distance from one corner to the opposite corner?
Answer: 20 feet
Step-by-step explanation: To find the diagonal distance from one corner to the opposite corner of Bernard's rectangular bedroom, we can use the Pythagorean theorem, which states that the square of the length of the hypotenuse (diagonal) of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
In this case, the two other sides are the length and the width of the room, so we have:
diagonal^2 = 12^2 + 16^2
diagonal^2 = 144 + 256
diagonal^2 = 400
Taking the square root of both sides, we get:
diagonal = √400
diagonal = 20 feet
Therefore, the diagonal distance from one corner to the opposite corner of Bernard's rectangular bedroom is 20 feet.
a hummingbird lives in a nest that is 5 meters high in a tree. the hummingbird flies 9 meters to get from its nest to a flower on the ground. how far is the flower from the base of the tree? if necessary, round to the nearest tenth.
The flower is about 10.3 meters away from the base of the tree, rounded to the nearest tenth of a meter. The hummingbird has to fly this distance to get to the flower.
To figure out how far the flower is from the base of the tree, we need to use the Pythagorean theorem. It can be applied when there is a right triangle, which is a triangle with one angle of 90 degrees.Here, the hummingbird's nest is at the top of the tree, and the flower is on the ground. The vertical distance from the nest to the ground is 5 meters. The horizontal distance from the tree trunk to the flower is the distance we want to find.
We'll need to calculate the length of the hypotenuse (the slanted line) of the right triangle in order to determine the distance from the tree to the flower. The hypotenuse's length is found by squaring each of the other sides, adding the results together, and then taking the square root:
hypotenuse=√5^2+9^2
=√25+81 = √106
≈10.3 m
So the flower is about 10.3 meters away from the base of the tree, rounded to the nearest tenth of a meter. The hummingbird has to fly this distance to get to the flower.
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The standard deviation of the weights of elephants is known to be approximately 15 pounds. We wish to construct a 95% confidence interval for the mean weight of newborn elephant calves. Fifty newborn elephants are weighed. The sample mean is 244 pounds. The sample standard deviation is 11 pounds Construct a 95% confidence interval for the population mean weight of newborn elephants. State the confidence interval (Round your answers to two decimal places.) Sketch the graph. (Round your answers to two decimal places.) CL - 0.95 X Calculate the error bound (Round your answer to two decimal places)
The error bound for the 95% confidence interval is (1.96 x Standard Deviation/√n), which in this case is (1.96 x 11/√50) = 2.56. This means that the true mean weight of newborn elephant calves lies within +/-2.56 pounds of the interval range.
The 95% confidence interval for the population mean weight of newborn elephants can be calculated using the sample mean of 244 pounds and the sample standard deviation of 11 pounds. The confidence interval is calculated using the following formula:
Confidence Interval = (Mean - (1.96 x Standard Deviation/√n)), (Mean + (1.96 x Standard Deviation/√n))
Where n is the sample size.
Therefore, the 95% confidence interval for the population mean weight of newborn elephants is (231.14, 256.86).
This can also be represented in a graph. The graph would have the x-axis representing the confidence interval, with a range from 231.14 to 256.86, and the y-axis representing the probability, which would be 0.95.
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what is the vertex of h=-16t^2+29t+6 and its domain and range, and x and y axis?
Given triangle AEB and triangle DFC ,side ABCD
As we have prove that the triangle has ΔEAB is congruent to ΔFDC.
Next, we can use the fact that AC = DB to prove that ΔEAB and ΔFDC have a pair of congruent sides. Specifically, since AC = DB and AE is parallel to DF, we know that triangles ACD and BDF are congruent by the Side-Angle-Side (SAS) congruence theorem. Therefore, we can conclude that AD = BC and CD = BD.
Now we can use the congruent angles and sides to prove that the remaining sides and angles of ΔEAB and ΔFDC are congruent. Specifically, we know that ∠AEB is congruent to ∠FDC and ∠EAB is congruent to ∠FDC because of the angle congruence we established earlier.
Additionally, we know that AB is congruent to CD and AD is congruent to BC because of the side congruence we established earlier. Finally, we know that AC = DB because this was given in the problem statement.
By using these angle and side congruences, we have shown that ΔEAB and ΔFDC are congruent by the Side-Angle-Side (SAS) congruence theorem.
Therefore, we have proven that ΔEAB is congruent to ΔFDC.
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Complete Question:
Given: ΔAEB and ΔDFC, ABCD, AE || DF, EB || FC, AC = DB
Prove: ΔEAB ≅ ΔFDC
The tape diagram represents an equation.
Write an equation to represent the image.
Answer: y+y=7
Step-by-step explanation:
7 is as big as 2 y's therefore y+y would be equal to 7
prove that n lines separate the plane into (n2 n 2)/2 regions if no two of these lines are parallel and no three pass through a common point.
In mathematical induction, prove that n lines separate the plane into (n^2 + n + 2)/2 regions if no two of these lines are parallel and no three pass through a common point.
Mathematical induction is a mathematical proof method that is used to demonstrate a statement or formula for all values of n, where n is a positive integer. If we use induction, we can show that the formula is true for n = 1, and we can also show that if the formula is true for n = k, then it is also true for n = k + 1.
Proof for n lines separating the plane into (n^2 + n + 2)/2 regions is given below:
Base Case: The theorem is true for n = 1. When we draw one line in the plane, we see that it splits the plane into two regions. Hence the formula is true for n = 1.
Induction Hypothesis: We believe that the formula is true for k lines. That is, k lines split the plane into (k^2 + k + 2)/2 regions.
Induction Step: We want to demonstrate that the formula is also true for k + 1 lines. We first take an arbitrary line from these k + 1 lines, which we call l. We notice that this line splits the plane into two regions.
Now, for the remaining k lines, we make an induction argument. We are sure that the formula is true for k lines. Thus, the k lines split the plane into (k^2 + k + 2)/2 regions. We know that these k lines intersect the line l at k points. Thus, by adding line l, we create k + 1 regions on the plane between these lines.
We now consider the line l itself. It can't cross any of the other k lines, or it would not meet our requirements. Therefore, it crosses each of the k existing lines, generating k + 1 areas. Thus, with the inclusion of line l, the number of regions on the plane is (k^2 + k + 2)/2 + k + 1 = (k^2 + 3k + 4)/2.
The formula for k + 1 is (k + 1)^2 + (k + 1) + 2 = k^2 + 3k + 4, and it is thus identical to the formula for the number of regions when k lines are drawn on the plane.
Therefore, the statement is true for all positive integers n.
Therefore, we have proved that n lines separate the plane into (n^2 + n + 2)/2 regions if no two of these lines are parallel and no three pass through a common point, by using mathematical induction.
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Find a vector equation and parametric equations in tfor the line through the point and parallel to the given line.(P0 corresponds to t = 0.)
P0 = (0,12, -10)
x = -4 + 2t, y = 7 - 4t, z = 5 + 8t
How do you find x,y,and z?
The vector equation and the parametric equations in t for the line through the point and parallel to the given line are:
Vector Equation= [-4 7 5] + t[2 -4 8]Parametric Equations:x= 2t - 4
y= -4t + 7
z= 8t + 5
How to find the value of x, y, and zTo find x, y, and z in the given scenario, the following steps can be followed:
1: Vector Equation of Line
To find the vector equation, use the given line and its coefficients:
x = -4 + 2t
y = 7 - 4t
z = 5 + 8t
Take the coefficients of x, y, and z, and place them in a 3 by 1 matrix:
Column Matrix= [-4 7 5]
Add the parameter t and place it in a column matrix to get the vector equation:
Vector Equation= [-4 7 5] + t[2 -4 8]
2: Parametric Equation.
To find the parametric equations, write the components of the vector equation in terms of the parameters:
x= -4 + 2t
y= 7 - 4t
z= 5 + 8t
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Eric is adding water to a 60 -gallons pool. The pool already has 12 gallons of water, and he wants to fill it to at least 27 gallons. The water flows at a rate of 6 gallons per minute. How many minutes, x , will it take for Eric to fill the pool with at least 27 gallons of water?
Which solution represents the answer to the problem solution, and which represents the solution for the inequality
It will take Eric 2.5 minutes to add 15 gallons of water to the pool and fill it to at least 27 gallons.
To fill the pool with at least 27 gallons of water, Eric needs to add:
27 - 12 = 15 gallons of water
The rate at which the water flows is 6 gallons per minute, so we can set up the equation:
6x = 15
where x is the number of minutes it will take for Eric to add 15 gallons of water to the pool.
Solving for x, we get:
x = 15/6
x = 2.5
This is the inequality that represents the situation because it shows that the amount of time Eric needs to add water to the pool must be greater than or equal to 15/6 minutes (or 2.5 minutes), which is the minimum time required to add 15 gallons of water.
The gallon has been in use since at least the 14th century and has undergone several changes in its size and definition over the centuries. It is commonly used in the United States, United Kingdom, and other countries that use the imperial system of measurement. One gallon is equal to 3.785 liters, and it is divided into four quarts or eight pints. In the US, a gallon is often used to measure the volume of gasoline, milk, and other liquids.
The word "gallon" has its roots in the Old French word "galon," which meant "measure of liquid." In the US, there are two different types of gallons: the US gallon, which is based on the Winchester gallon used in the late 18th century, and the imperial gallon, which is used in the UK and other Commonwealth countries.
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3. The length of one leg of a 45-45-90 triangle is 7 m. What is the length of the other leg and the length of the hypotenuse?
The other leg is 7 m, and the hypotenuse is 7 m.
O The other leg is 7 m, and the hypotenuse is 14 m.
O The other leg is 7√2 m, and the hypotenuse is 7 m.
The other leg is 7 m, and the hypotenuse is 7√2 m.
the length of the other leg is 7m and the length of the hypotenuse is 7√2 m.
Pythagoras Theorem StatementIn the right-angled triangle, the square of the hypotenuse side is equals to the sum of the squares of the other two sides, according to Pythagoras's Theorem. This triangle's three sides are known as the Perpendicular, Base, and Hypotenuse. Because to its position opposite the 90° angle, the hypotenuse in this case is the longest side.
The definition yields the following as the Pythagoras Theorem formula:
Hypotenuse² = Perpendicular² + Base²
c² = a² + b²
Length of one leg=7m
Angles of triangle are 45°,45° and 90°
According to Pythagoras theorem,
x²=7²+7²
x=7√2
the length of the other leg is 7m and the length of the hypotenuse is 7√2 m.
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What are the answers for all the blanks?
Answer:
The graph is a dotted line from ( 0, 12 ) to ( 15 , 0 )
The shaded region is above the boundary line. The origin is not in the shaded region.
===================================================
Explanation:
x = number of shirts
y = number of pants
16x = amount made from the shirts only
20y = amount made from the pants only
16x+20y = total amount made, aka revenue
16x+20y > 240
This is because Giselle needs to make more than $240 to be profitable.
------------------
The graph is a dotted line because the "or equal to" is not part of the inequality sign. If Giselle could make $240 and be profitable, then we would use a solid line instead and use "or equal to". But in this case, she must make above $240.
Let's consider the boundary line 16x+20y = 240. Plug in x = 0 to get
16x+20y = 240
16*0+20y = 240
20y = 240
y = 240/20
y = 12
Therefore we can say (0,12) is one point on the dotted boundary line. It is the y intercept.
Use similar steps for y = 0 to find x.
16x+20y = 240
16x+20*0 = 240
16x = 240
x = 240/16
x = 15
The x intercept is (15,0) where the dotted line crosses the x axis.
Therefore, the dotted boundary line goes through (0,12) and (15,0).
------------------
Now to the question: where to shade?
Let's check the origin point (0,0). Meaning we plug in x = 0 and y = 0.
16x+20y > 240
16*0+20*0 > 240
0+0 > 240
0 > 240
Clearly that's false so (0,0) is NOT in the shaded solution region. We shade the opposite region of the origin. We'll shade above the boundary as indicated in the diagram below. I used GeoGebra to make the graph. Desmos is another good option.
I need help pls help me find the area:
Answer:
Step-by-step explanation:
348.55
Find the area of the figure.
Answer:
A = 32 ft²
Step-by-step explanation:
the area (A ) of a square is calculated as
A = s² ( s is the side length )
the diagonal divides the square into 2 right triangles
using Pythagoras' identity on the lower right triangle with hypotenuse 8 and sides s , then
s² + s² = 8²
2s² = 64 ( divide both sides by 2 )
s² = 32
Then
A = s² = 32 ft²
Lee buys 12 notebooks for 1. 29 each. How much money does lee spend on the 12 notebooks
Lee buys 12 notebooks and the cost of each one is $1.29. The total cost of all twenty notebooks is equals to $15.48. So, Total $15.48, money lee spends on the 12 notebooks.
We have, Lee buys some notebooks by spending money.
Number of notebooks that she bought = 12
The price or cost of one notebook = $1.29
We have to determine the amount of money she spends on the 12 notebooks, that total cost of 12 notebooks. Let the total cost of 12 notebooks be 'x dollars'. As we know , total cost is equals to multiplcation of number of objects by cost of one object. So, total cost of 12 notebooks = number of notebooks × cost of one notebook
=> total cost of 12 notebooks = 12 ×1.29
= (12×129)/100
= 15.48
Hence, the required cost is $15.48.
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Complete question:
Lee buys 12 notebooks for $1.29 each. How much money does lee spend on the 12 notebooks.
5 2 fiths minus 1 2 fiths
Answer:
Step-by-step explanation:
2/5-1 2/5
PLS ANSWER THIS ASAP
In two similar triangles, the ratio of the lengths of a pair of corresponding sides is 7:8. If the perimeter of the larger triangle is 32, find the perimeter of the smaller triangle.
The perimeter of the smaller triangle would be = 28.1
How to calculate the perimeter of the smaller triangle?A triangle can be defined as a three sided polygon that has a total internal angle of 180°.
To calculate the perimeter of the triangle is to find out the scale factor that exists between the two triangles.
The formula for scale factor = original object/new object
Scale factor= 8/7 = 1.14
The perimeter of the smaller triangle = 32/1.14
= 28.1.
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an inner city revitalization zone is a rectangle that is twice as long as it is wide. the width of the region is growing at a rate of 32 m per year at a time when the region is 220 m wide. how fast is the area changing at that point in time?
The area is changing at a rate of 28,160 m²/year at that point in time.
The area of the rectangular region is given by:
A = lw
Where l is the length of the rectangular region and w is the width of the rectangular region.
The width of the rectangular region is given to be 220 m. Therefore, we have the width w = 220 m. The length l of the rectangular region can be found knowing that it is twice as long as it is wide. Therefore, the length of the rectangular region is given by:
l = 2w
l = 2 x 220
l = 440
Therefore, the length l of the rectangular region is 440 m.
At the given point in time, the width of the rectangular region is growing at a rate of 32 m per year. Therefore, we have the rate of change of the width dw/dt to be 32 m per year. We need to find how fast the area of the rectangular region is changing at that point in time. Therefore, we need to find the rate of change of the area of the rectangular region dA/dt.
A = lw
dA/dt = w dl/dt + l dw/dt
dA/dt = 220 d/dt(2w) + 440 dw/dt
dA/dt = 220 x 2 dw/dt + 440 dw/dt
dA/dt = 880 dw/dt
Substitute the value of dw/dt to get:
dA/dt = 880 x 32
dA/dt = 28,160 m²/year
Therefore, the area of the rectangular region has a rate of change of 28,160 m² per year at that point in time.
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Based on the following sorted 20 values for age, what are the possible split points?
{20, 22, 24, 26, 28, 31, 32, 33, 35, 40, 42, 43, 45, 47, 49, 50, 52, 53, 55, 57}
Multiple Choice
a {20, 21, 23, 25, 27, 29. 5, 31. 5, 32. 5, 34, 37. 5, 41, 42. 5, 44, 46, 48, 49. 5, 51, 52, 54, 56}
b {21, 23, 25, 27, 29. 5, 31. 5, 32. 5, 34, 37. 5, 41, 42. 5, 44, 46, 48, 49, 51, 52. 5, 54, 56, 57}
c {0, 21, 23, 25, 27, 29. 5, 31. 5, 32. 5, 34, 37. 5, 41, 42. 5, 44, 46, 48, 49, 51, 52. 5, 54, 56}
d {21, 23, 25, 27, 29. 5, 31. 5, 32. 5, 34, 37. 5, 41, 42. 5, 44, 46, 48, 49. 5, 51, 52. 5, 54, 56}
Based on the following sorted 20 values for age, the possible split points are {20, 21, 23, 25, 27, 29. 5, 31. 5, 32. 5, 34, 37. 5, 41, 42. 5, 44, 46, 48, 49. 5, 51, 52, 54, 56} (option a).
Option A suggests that the split points are {20, 21, 23, 25, 27, 29.5, 31.5, 32.5, 34, 37.5, 41, 42.5, 44, 46, 48, 49.5, 51, 52, 54, 56}. Notice that every split point falls between two consecutive ages in the original list. For example, the first split point is 20 because it is between 20 and 22. The second split point is 21 because it is between 20 and 22 as well.
Option B suggests that the split points are {21, 23, 25, 27, 29.5, 31.5, 32.5, 34, 37.5, 41, 42.5, 44, 46, 48, 49, 51, 52.5, 54, 56, 57}. Notice that the only difference between this option and Option A is that the last split point is 57 instead of 49.5.
Option C suggests that the split points are {0, 21, 23, 25, 27, 29.5, 31.5, 32.5, 34, 37.5, 41, 42.5, 44, 46, 48, 49, 51, 52.5, 54, 56}. Notice that the first split point is 0, which is not a possible age in the original list.
Option D suggests that the split points are {21, 23, 25, 27, 29.5, 31.5, 32.5, 34, 37.5, 41, 42.5, 44, 46, 48, 49.5, 51, 52.5, 54, 56}. Notice that the only difference between this option and Option A is that the split point after 49 is 49.5 instead of 49.5.
In summary, the correct answer is Option A because it provides all the possible split points that fall between the ages in the original list. When working with split points, it's important to consider the specific context and criteria for dividing the data.
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