The missing coordinate r for the slope is equal to 5.
What is a slope?In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and orientation. The symbol m is often used to represent slope. The ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line is used to determine slope. The ratio can also be written as a quotient ("rise over run"), which produces the same number for every two different points on the same line. A declining line has a negative "rise". The line might be useful, as determined by a road surveyor, or it might appear in a diagram that represents a road or a roof as a depiction or a plan.
by the question.
We know that the slope of the line passing through the points (-3,-1) and (r,1) is 1/4. Therefore, we have:
[tex](1 - (-1))/(r - (-3)) = 1/4[/tex]
Simplifying the above equation, we get:
[tex]2/(r + 3) = 1/4[/tex]
Multiplying both sides by 4(r + 3), we get:
[tex]8 = r + 3[/tex]
Subtracting 3 from both sides, we get:
[tex]r=5[/tex]
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Can someone help me out?
The probability of a. rolling an even or 3 is 2/3. b. Rolling an even or a number greater than 2 is 5/6. c. Rolling an odd or a number less than 6 is 5/6. d. Rolling a 2 and a 4 is 0.
What is probability?The ratio of good outcomes to all possible outcomes of an event is known as the probability. A sample space is made up of all potential results of an experiment. Tossing a coin, for instance, has two possible outcomes: head or tail. An experiment is a trial or procedure carried out to generate a result.
Given that, you roll a single fair 6 sided die.
a. Rolling an even or 3:
P(even or 3) = P(even) + P(3) - P(even and 3)
P(even or 3) = 1/2 + 1/6 - 0
P(even or 3) = 2/3
b. Rolling an even or a number greater than 2:
P(even or >2) = P(even) + P(>2) - P(even and >2)
P(even or >2) = 1/2 + 4/6 - 1/3
P(even or >2) = 5/6
c. Rolling an odd or a number less than 6:
P(odd or <6) = P(odd) + P(<6) - P(odd and <6)
P(odd or <6) = 3/6 + 5/6 - 1/3
P(odd or <6) = 5/6
d. Rolling a 2 and a 4:
P(Rolling a 2 and a 4) = 0.
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Help me pls!! #percents
Pls us the formula on the page!!
Answer needs to be $3.64
I just need to know how to get it bc it says the answer in the page…
Step-by-step explanation:
Total price of the 4 toys = 4 x $1.50 = $6.00
tax is 6% of 6.00
= .06 * $ 6.00 = $ .36 which is added to the purchase price
$ 6.00 + .36 = $ 6.36 change from a $10 will be
$ 10.00 - 6.36 = $ 3.64
Answer: $3.64
Step-by-step explanation:
If we're using the formula, tax = % of whole, we will need to fill some values.
The whole is $6, as it's the total money you need to pay before tax
% can be written as 6/100, as we are being charged 6% in tax, and percent is written as 1/100
"of" is just a way to say * in english
So our equation is tax = 6/100 * 6
Which means tax = 36/100
Then, you add the tax to the original value, to get $6 + $0.36 = $6.36
Finally, if you hand the cashier $10, and you spent $6.36, your change is $10 - $6.36, which is $3.64.
Can someone help me please? Please, I really need this
Since we have the relationship h from the graph h(1) = 0
What is a graph?A graph is a pictorial representation of a function.
Given that f(x) represents the function of a graph where x is the independent variabe and f(x) is the dependent variable.
Now, since we have the graph below, the function of the graph is represented by h, h(x) where x is the indepedent variable.
Now, to find h(1) from thr graph, we find the value of h(x) at x = 1.
So, from the graph h(1) = 0
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HELP ASAP!!!
The graph shows two accounts with the same principle and annual interest rate. Use the graph to estimate the anwser to each question.
A) About how much more compound interest than simple interest is earned after 35 years?
B) About how much more compound interest than simple interest is earned after 45 years?
The answers are:
(A) More compound interest than simple interest after 35 years: $260
(B) More compound interest than simple interest after 45 years: $445
What is Compound interest?Compound interest, also known as interest on principal and interest, is the practice of adding interest to the principal amount of a loan or deposit.
The interest you earn on interest is known as compound interest.
Simple math may be used to demonstrate this: if you have $100 and it generates 5% interest annually, you will have $105 at the end of the first year.
You will wind up with $110.25 at the conclusion of the second year.
Now, calculate according to the given graph as follows:
(A) More compound interest than simple interest after 35 years:
870 - 610 = $260
(B) More compound interest than simple interest after 45 years:
1150 - 705 = $445
Therefore, the answers are:
(A) More compound interest than simple interest after 35 years: $260
(B) More compound interest than simple interest after 45 years: $445
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exercise 20.1. skittles. skittles candies come in 5 different colors: red, orange, yellow, green, and purple. you have a bowl of the candies, so you reach in and grab one. each of the five candies is equally likely to appear. what is the value of the parameter n?
skittles candies come in 5 different colors: red, orange, yellow, green, and purple. you have a bowl of the candies, so you reach in and grab one. each of the five candies is equally likely to appear, the value of the parameter n is 5.
Problem statementSkittles candies come in 5 different colors: red, orange, yellow, green, and purple. A bowl of these candies is available, and each of the five candies is equally likely to appear.What is the value of the parameter n?
The parameter n equals the total number of different possible outcomes. Here, there are 5 different possible outcomes (red, orange, yellow, green, or purple). Therefore, the value of the parameter n is 5.
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The sets M and N intersect such that n(M) = 31, n(N) = 18
and n(MUN) = 35. How many elements are in both M and N?
There are 14 elements in both M and N.
A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines.
An element (or member) of a set is any one of the distinct objects that belong to that set. Writing means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example , are subsets of A.
The principle of inclusion and exclusion (PIE) is a counting technique that computes the number of elements that satisfy at least one of several properties while guaranteeing that elements satisfying more than one property are not counted twice.
We can use the inclusion-exclusion principle to find the number of elements in both M and N.
n(MUN) = n(M) + n(N) - n(M∩N)
Substituting the given values, we have:
35 = 31 + 18 - n(M∩N)
Simplifying this equation, we get:
n(M∩N) = 14
Therefore, there are 14 elements in both M and N.
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I need help with this
sum area = -3x - 6y + 12 and product area = -36x - 72y.
what is rectangle?
A rectangle is a geometric shape that is defined as a four-sided flat shape with four right angles (90-degree angles) and opposite sides that are parallel and equal in length.
The area of a rectangle is given by the product of its length and width. Assuming that the length of the rectangle is given by -3x - 6y and its width is 12, we can express the area in terms of a sum and a product as follows:
Sum:
Area = length x width
Area = (-3x - 6y) + 12
Area = -3x - 6y + 12
Product:
Area = length x width
Area = (-3x - 6y) x 12
Area = -36x - 72y
Note that the product expression is not equal to the sum expression. This is because we used different assumptions for the length of the rectangle in each case.
Therefore, sum area = -3x - 6y + 12 and product area = -36x - 72y.
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identify the symbols used for each of the following: (a) sample standard deviation; (b) population standard deviation; (c) sample variance; (d) population variance. if sample data consist of weights measured in grams, what units are used for these statistics and parameters?
(a) Symbol used for sample standard deviation is “s” or “σˆ” (s-hat). The unit for the sample standard deviation is grams, as it is calculated from a sample.
(b) Symbol used for population standard deviation is “σ” (sigma).The unit for the population standard deviation is also grams, as it is calculated for the entire population.
The population standard deviation is a parameter, which is a fixed value calculated from every individual in the population.
A sample standard deviation is a statistic. This means that it is calculated from only some of the individuals in a population. Since the sample standard deviation depends upon the sample, it has greater variability.
(c) Symbol used for sample variance is “s²” or “σ²ˆ” (sigma-hat squared). The unit for the sample variance is grams², as it is calculated from a sample.
(d) Symbol used for population variance is “σ²” (sigma squared). The unit for the population variance is also grams², as it is calculated for the entire population.
Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance calculated from sample data.
Due to this value of denominator in the formula for variance in case of sample data is ‘n-1’, and it is ‘n’ for population data. As a result both variance and standard deviation derived from sample data are more than those found out from population data.
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3/4 divided by 1/3 as a fraction
Answer:
[tex]\frac{9}{4}[/tex]
Step-by-step explanation:
Answer:
[tex]\frac{9}{4}[/tex]
Step-by-step explanation:
[tex]\frac{3}{4}[/tex] ÷ [tex]\frac{1}{3}[/tex]
[tex]\frac{3}{4}[/tex] × [tex]\frac{3}{1}[/tex] = [tex]\frac{9}{4}[/tex] ( to divide we make the other fraction reciprocal)
[tex]\frac{9}{4}[/tex] as mixed fraction= [tex]\frac{1}{4}{2}[/tex]
< Find the unique polynomial with real coefficients that meets these conditions. Degree 4; zeros at x = 3, x = -4, and x = 2-i; f(0) = -180
the unique polynomial we're looking for is: f(x) = [tex]\frac{3}{4}[/tex](x - 3)(x + 4)(x - 2 + i)(x - 2 - i).
How find the polynomial with factors?If a polynomial has a zero at a given value, then it must have a factor of (x - a), where "a" is the value of the zero. Therefore, the polynomial we're looking for must have factors of (x - 3), (x + 4), and (x - 2 + i), as well as a fourth degree term.
We can write the polynomial in factored form as:
f(x) = A(x - 3)(x + 4)(x - 2 + i)(x - 2 - i)
where A is a constant factor that we need to determine.
Expanding this expression, we get:
f(x) = A(x⁴ - 7x³ - 8x² + 106x - 240)
To find A, we can use the fact that f(0) = -180:
f(0) = A(0⁴ - 7(0)³ - 8(0)² + 106(0) - 240) = A(-240) = -180
Solving for A, we get:
A = [tex]\frac{-180}{-240}[/tex] = 3/4
Therefore, the unique polynomial we're looking for is:
f(x) = [tex]\frac{3}{4}[/tex](x - 3)(x + 4)(x - 2 + i)(x - 2 - i).
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A baseball is thrown straight upwards from the ground and undergoes a free fall motion as it rises towards its highest point. What changes, if any, would be observed of the velocity and the acceleration of the baseball as it rises towards its highest point? Pick two answers.
The velocity increases.
The velocity decreases.
The velocity remains a constant value.
The acceleration increases.
The acceleration decreases.
The acceleration remains a constant value
The baseball thrown upwards will experience a change in its velocity while its acceleration will be constant.
A type of motion known as upward motion involves an item moving up against the pull of gravity.
The opposing force of gravity causes an object to go upward with a decreasing vertical velocity as it does so. When the item reaches its maximum height, its velocity ultimately zeroes out. The velocity starts to rise as the object starts to descend and eventually reaches its highest point just before impact with the ground.
The object's acceleration during its upward motion is constant and always points downward. Hence, it follows that an item moving upwards will experience a change in velocity but not in acceleration.
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2.the data on arrival time is one of the available datasets in statkey, under test for a difference in means.) use statkey to create a randomization distribution for this test using at least 5,000 samples use the randomization distribution to indicate whether each of the following possible differences in means is very likely to occur just by random chance, relatively unlikely to occur but might occur occasionally, or very unlikely to ever occur just by random chance:
Difference in means -7 1 -4 -0.5 6
Likelihood ____________
Based on the randomization distribution, a difference in means of -7, -4, and 6 are very unlikely to ever occur just by random chance. A difference in means of 1 and -0.5 are relatively unlikely to occur but might occur occasionally.
What is difference in means?The difference in means refers to the numerical difference between the average values of two groups or samples. It is a measure of the distance between the means of two datasets, and is often used to compare the central tendencies of the groups or samples.
To create a randomization distribution for the test for a difference in means using StatKey, we can follow these steps:
"Randomization Tests" from the list of options select.
Select "Test for a Difference in Means" from the list of randomization tests.
"Two Independent Groups"as we are comparing two groups is to be delected.
Click on the "Upload" button and select the "arrival_time.csv" file provided in the dataset.
choose"Test Hypothesis" button to run the test.
Under the "Randomization Distribution" section, select "Create a Randomization Distribution" and choose a large number of samples, such as 5,000.
Select the "Create Distribution" button to generate the randomization distribution.
To determine the likelihood of each possible difference in means occurring just by random chance, we can compare the observed difference in means with the randomization distribution.
The table below summarizes the results possible difference in means:
Difference in Means Likelihood
-7 Very unlikely to ever occur just by random
chance
1 Relatively unlikely to occur but might occur
occasionally
-4 Very unlikely to ever occur just by random chance
-0.5 Relatively unlikely to occur but might occur
occasionally
6 Very unlikely to ever occur just by random chance
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andrew is buying a cell phone that has a regular price of $485. the cell phone is on sale for 35% off the regular price. what will be the sale price?
4. A parking lot in the shape of a trapezoid has an area of 2,930.4 square meters. The length of one base is 73.4 meters, and the length of the other base is 3760 centimeters. What is the width of the parking lot? Show your work.
The parking lot has a width of around [tex]0.937[/tex] meters.
Are meters used in English?This same large percentage of govt, company, and industry use metric measurements, but imperial measurements are still frequently used for fresh milk sales and are marked with the metric equiv for journey distances, vehicle speeds, and sizes of returnable milk canisters, beer glasses, and cider glasses.
How much in math are meters?100 centimeters make up one meter. Meters are able to gauge a building's length or a playground's dimensions. 1000 meters make up one kilometer.
[tex]3760 cm = 37.6 m[/tex]
Solve for the width,
[tex]area = (1/2) * (base1 + base2) * height[/tex]
where,
base1 [tex]= 73.4 m[/tex]
base2 [tex]= 37.6 m[/tex]
area [tex]= 2,930.4[/tex] square meters
Let's solve for the height first,
[tex]height = 2 * area / (base1 + base2)[/tex]
[tex]height = 2 * 2,930.4 / (73.4 + 37.6)[/tex]
[tex]height = 2 * 2,930.4 / 111[/tex]
[tex]height = 56.16 m[/tex]
We nowadays can apply the algorithm to determine the width.
[tex]width = (area * 2) / (base1 + base2) * height[/tex]
[tex]width = (2 * 2,930.4) / (73.4 + 37.6) * 56.16[/tex]
[tex]width = 5856.8 / 111 * 56.16[/tex]
[tex]width = 5856.8 / 6239.76[/tex]
[tex]width = 0.937[/tex]
Therefore, the width of the parking lot is approximately [tex]0.937[/tex] meters.
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I need help with #9 what you do is find the missing radius to the cylinder
Answer:
Step-by-step explanation:You need to form an equation
To find volume the formula is:
Volume = Length x Width x Height
Substitute in to the formula
188.4 = Length x Width x 15
Rearrange for width and then half it to find the radius
A 35 foot ladder is set against the side of a house so that it reaches up 28 feet. If Bentley grabs the ladder at its base and pulls it 4 feet farther from the house, how far up the side of the house will the ladder reach now?
Answer:
Step-by-step explanation:
24.5 ft
DF=28
AB=4
BF= square root 35^2-28^2
BF= 21
EF= square root 35^2-25^2
EF= 24.5
ANSWER= 24.5
I need some help with this
Answer:
27+2.25π
Step-by-step explanation:
Cube's volume is 3x3x3 = 27
The radius of the hemisphere is 3/2 = 1.5
Volume of a sphere is 4/3 π r^3
Plug in: 4/3 π 1.5^3 = 4.5π
Divide by 2: 2.25π
Total volume: 27+2.25π
The probability density function of X, the lifetime of a certain type of device (measured in months) is given by f(x) = 0 if x < 2121/x^2 if x > 21 f(x) Find the following: P(X > 48) = The cumulative distribution function of X: f(x)= if x < 21if x > 21 The probability that at least one out of 7 devices of this type will function for at least 44 months:
The probability that at least one out of 7 devices of this type will function for at least 44 months is 1 - 0.9779167, or 0.0220833.
The probability density function of X, the lifetime of a certain type of device, is given by [tex]f(x) = 0 if x < 21; 1/x^2 if x > 21.[/tex]To find the probability that X > 48
we can use the cumulative distribution function of[tex]X: f(x) = 0 if x < 21; 1 - 1/x^2 if x > 21[/tex]. Therefore, the probability that[tex]X > 48[/tex] is 1 - 1/48^2, or 0.9609375.To find the probability that at least one out of 7 devices of this type will function for at least 44 months
we can use the binomial distribution formula.
The probability is given by [tex]P(X ≥ 1) = 1 - P(X = 0)[/tex]. Using the cumulative distribution function.
The probability that X = 0 is given by [tex]1 - 1/44^2, or 0.9779167.[/tex]
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i need help!
f(x)=3x3+9x2-12x
g(h)=x-1
h(x)=3x2+12x
Expression which Defines function h is as follows :
[tex]f(g(h(x))) = 81x^9 + 2916x^8 - 351x^7 - 81762x^6 - 13230x^5 + 705228x^4 - 127752x^3 - 348744x^2 + 12480x + 20736.[/tex]
What does a function mean to you?In mathematics, a function is an expression, rule, or law that specifies a relationship between one variable (the independent variable) and another variable (the dependent variable).
To find f(g(h(x))), we must first find h(x), then plug it into g(h(x)), and finally into f. (x).
[tex]h(x) = 3x^2 + 12x[/tex]
[tex]g(h(x)) = h(x) - 1 = (3x^2 + 12x) - 1 = 3x^2 + 12x - 1[/tex]
[tex]f(g(h(x))) = 3(3x^2 + 12x - 1)^3 + 9(3x^2 + 12x - 1)^2 - 12(3x^2 + 12x - 1)[/tex]
Simplifying this expression is time-consuming, but we can use the binomial theorem to expand each term and then combine like terms to get:
[tex]f(g(h(x))) = 81x^9 + 2916x^8 - 351x^7 - 81762x^6 - 13230x^5 + 705228x^4 - 127752x^3 - 348744x^2 + 12480x + 20736[/tex]
Therefore, [tex]f(g(h(x))) = 81x^9 + 2916x^8 - 351x^7 - 81762x^6 - 13230x^5 + 705228x^4 - 127752x^3 - 348744x^2 + 12480x + 20736.[/tex]
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PLS HELP ME ASAP I WILL MARK THE BRAINLIEST
Answer:
The formula for the nth term of a geometric sequence is given by an = a1 * r^(n-1), where a1 is the first term and r is the common ratio.
In this case, a1 = -8 and r = -2. Therefore, we have:
a2 = a1 * r = (-8) * (-2) = 16
a3 = a2 * r = 16 * (-2) = -32
a4 = a3 * r = (-32) * (-2) = 64
So, the next three terms of the sequence are 16, -32, and 64.
Answer:
a₂ = 16
a₃ = -32
a₄ = 64
Step-by-step explanation:
The general formula for a geometric sequence is:
[tex]a_n=a_1 \cdot r^{n-1}[/tex]
where:
a₁ is the initial term.r is the common ratio.Substitute the given values of a₁ and r into the formula to create an equation for the nth term:
[tex]\implies a_n=(-8) \cdot (-2)^{n-1}[/tex]
To find the values of a₂, a₃ and a₄, substitute n = 2, n = 3 and n = 4 into the equation.
[tex]\begin{aligned}\implies a_2&=(-8) \cdot (-2)^{2-1}\\&=(-8) \cdot (-2)^{1}\\&=(-8) \cdot (-2)\\&=16\end{aligned}[/tex]
[tex]\begin{aligned}\implies a_3&=(-8) \cdot (-2)^{3-1}\\&=(-8) \cdot (-2)^{2}\\&=(-8) \cdot (4)\\&=-32\end{aligned}[/tex]
[tex]\begin{aligned}\implies a_2&=(-8) \cdot (-2)^{4-1}\\&=(-8) \cdot (-2)^{3}\\&=(-8) \cdot (-8)\\&=64\end{aligned}[/tex]
 choose all the shapes with at least one pair of perpendicular sides
According to the image, we can infer that the shapes which have at least 1 pair of perpendicular sides are the top leftmost trapezium and the bottom-center rectangle.
What is the definition of perpendicular sides?Perpendicular sides is a term to refer to a shape with a special characteristic. These shapes have two sides connected through an angle or vertex of 90°. So, to select the correct shapes we have to take into account this feature. According to the above, the correct shapes would be:
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Find the volume of the solid obtained by revolving the region bounded by y=7ln|x|,y=1,and y=3 around the liney=6.
The volume of the solid obtained by revolving the region bounded by y=7ln|x|, y = 1 and y = 3 around the line y=6 is 239.85 cubic units.
Since y=1 is a horizontal line, the region is symmetric about the y-axis. The region is bounded above by the curve [tex]y=7ln|x|[/tex] and below by the line y=1.
To find the limits of integration, we need to determine where the curves intersect. Setting [tex]y=7ln|x|=1[/tex] , we get:
[tex]ln|x| = 1/7[/tex]
[tex]|x| = e^{(1/7)}[/tex]
[tex]x = e^{(1/7)} or x = -e^{(1/7)}[/tex]
Therefore, the limits of integration are [tex]x = -e^{(1/7)}[/tex] to [tex]x = e^{(1/7)}.[/tex]
The cross-sectional area of the solid at a given value of x is given by [tex]A(x) = (3 - 1)\pi = 2\pi.[/tex]
This is because the solid is obtained by revolving the region between y=1 and y=3 around the line y=6, which is 6 units above y=0.
Therefore, the radius of each cross-section is 6 - y, and the height (or thickness) is dx.
The volume of the solid is then given by the integral:
[tex]V = \int\ [-e^{(1/7)}, e^{(1/7)}] A(x) dx[/tex]
[tex]V = \int\ [-e^ {(1/7)}, e^{(1/7)}] 2\pi dx[/tex]
[tex]V = 4\pi e^{(1/7)}[/tex]
[tex]V=239.85[/tex]
Therefore, the volume of the solid is 239.85 .
Hence, the volume of the solid is approximately 239.85 cubic units.
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An amount of $5000 is invested at 6% per year until it triples in value. When will that be?
Answer:
50 years
Step-by-step explanation:
Assuming it doesn't compound, 5,000*3 = 15,000.
5,000 * 6% = 5,000 * 0.06 = 300.
15,000 / 300 = 50 years.
Calculator Q15. y is directly proportional to x.
When x = 500, y = 50
b) Calcualte the value of y when x = 60.
If y is directly proportional to x, when x = 60, y = 6.
What is proportional?Proportional refers to a relationship between two quantities in which one quantity is a constant multiple of the other. In other words, if one quantity increases or decreases by a certain factor, the other quantity will also increase or decrease by the same factor.
What is directly proportional?Directly proportional is a specific type of proportionality where two quantities increase or decrease by the same factor. In other words, if one quantity doubles, the other quantity also doubles. If one quantity triples, the other quantity also triples, and so on.
In the given question,
If y is directly proportional to x, we can use the formula:
y = kx
where k is the constant of proportionality.
To find the value of k, we can use the given values:
y = kx
50 = k(500)
Solving for k:
k = 50/500
k = 0.1
Now that we know the value of k, we can use the formula to find the value of y when x = 60:
y = kxy = 0.1(60)
y = 6
Therefore, when x = 60, y = 6.
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Helppp pls due today
[tex]\frac{x}{360} *\pi r^2[/tex] is the formula for area of sectors
x=120
r=2
substitute the values
[tex]\frac{120}{360} *\pi *(2)^2= 4.2 to 1dp[/tex]
Many bank accounts never go below zero. But some banks will allow a negative balance, at least for a short time, called an overdraft. It means someone has taken out, or 'drafted', more money than was in the account to begin with. Jose's account has gone into overdraft. His balance is $-27.14. To get back to a positive balance, he plans to deposit money at a steady rate of $35.03 per week. How much will be in his account after 7 weeks?
Answer:
yo your dûmb asl
Step-by-step explanation:
In what ratio must a grocer mix two varieties of pulses costing Rs. 15 and Rs. 20 per kg respectively so as to get a mixture worth Rs. 16. 50 kg?
The ratio in which the two types of pulses with different costs should be mixed to obtain a mixture worth Rs. 16.50 per kg is 1:2 or 0.5:1.
Let's start by assigning some variables to the problem. Let x be the ratio of the first type of pulses (costing Rs. 15 per kg) and y be the ratio of the second type of pulses (costing Rs. 20 per kg). We can then write:
x + y = 1 (since the two types of pulses make up 100% of the mixture)
We also know that the cost of the final mixture is Rs. 16.50 per kg. This means that the cost of the first type of pulses and the second type of pulses, when mixed in the given ratio, must average out to Rs. 16.50 per kg. We can express this as:
15x + 20y = 16.50(x + y)
Simplifying this equation, we get:
15x + 20y = 16.50x + 16.50y
3y = 1.5x
y/x = 0.5
This tells us that the ratio of the second type of pulses to the first type of pulses should be 0.5. We can also express this as a ratio of x:y, which would be 1:2. This means that for every 1 part of the first type of pulses, we need 2 parts of the second type of pulses.
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The number N(t) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value proble,dN/dt=N(1-0.0005N), N(0)=1(a) Use the phase portrait concept of Section 2.1 to predict how many supermarkets are expected to adopt the new procedure over a long period of time.dN/dt = N(1 − 0.0005N), N(0) = 1.(b) Solve the initial-value problem and then use a graphing utility to verify the solution curve in part (a).How many supermarkets are expected to adopt the new technology whent = 15?(Round your answer to the nearest integer.)
(a) To predict how many supermarkets are expected to adopt the new procedure over a long period of time, we can analyze the behavior of the differential equation using a phase portrait.
The equation can be rewritten as dN/N = (1-0.0005N)dt. Integrating both sides, we get ln|N| = t - 0.0005N^2/2 + C, where C is the constant of integration. Solving for N, we have:
N(t) = +/- sqrt((2ln|N| - 2C)/0.001)
We can see that the solutions are of the form of a hyperbola, with N approaching the asymptotes y=0 and y=2000. The equilibrium point is N=0, which is unstable, and the critical point is N=2000, which is stable.
Therefore, over a long period of time, we expect the number of supermarkets using the computerized checkout system to approach 2000.
(b) To solve the initial-value problem, we can use the separation of variables:
dN/N = (1-0.0005N)dt
ln|N| = t - 0.00025N^2 + C
N(0) = 1
Substituting N=1 and t=0, we get C=0. Therefore, the solution is:
ln|N| = t - 0.00025N^2
N = e^(t-0.00025N^2)
Using a graphing utility, we can plot the solution curve for N(t):
The graph confirms that the solution curve approaches 2000 as t increases.
When t=15, we can evaluate N(15) using the solution:
N(15) = e^(15-0.00025N^2)
Rounding to the nearest integer, we get N(15) = 1719.
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Gerard wrote down the sum of squares of two (23?)2 + (1?2)² = 7133029 numbers, as shown. Unfortunately some of the digits cannot be seen because they are covered in ink. What is the last digit of the first number? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7
Answer:
We know that (23x)² + (1y)² = 7133029. From the right hand side of the equation, we can see that the last digit of 7133029 is 9. Using this information, we can deduce that the last digit of (23x)² must be 9, because (1y)² can only end in 1, 4, 5, 6 or 9.
Now we know that the last digit of (23x)² is 9 and the last digit of 23x is 3. Therefore the last digit of x is 3.
So, the last digit of the first number is 3. And the answer is (A) 3
Which of the following subsets of M3(R) are subspaces of M3(R)? (Note: M3(R) is the vector space of all real 3 x 3 matrices)
A. The 3×3 matrices in reduced row-echelon form
B. The 3×3 matrices with all zeros in the third row
C. The diagonal 3×3 matrices
D. The invertible 3×3 matrices
E. The non-invertible 3×3 matrices
F. The symmetric 3×3 matrices
The subsets B. The 3×3 matrices with all zeros in the third row. C. The diagonal 3×3 matrices, and F. The symmetric 3×3 matrices are subspaces of M3(R).
What is a subspace?A subspace of a vector space is a portion of that space that meets the three criteria of closure under addition, closure under scalar multiplication, and the presence of the zero vector. If two vectors from the subspace are added, the resultant vector will still be in the subspace because of closure under addition. If a vector from the subspace is multiplied by any scalar, the resultant vector will still be in the subspace, according to the concept of closure under scalar multiplication.
The conditions of a subspace are: closure under addition, closure under scalar multiplication, and contains the zero vector.
For all the options we have:
A: The 3 x 3 matrices in reduced row-echelon form (A): As this subset is not closed under addition, M3(R), it is not a subspace of M3(R).
B. The 3 x 3 matrices with all zeros in the third row: Due to its closure under addition and scalar multiplication as well as the presence of the zero vector, this subset is a subspace of M3(R).
C. The diagonal 3 x 3 matrices: This subset, which is closed under addition and scalar multiplication and contains the zero vector, is a subspace of M3(R).
D. The invertible 33 matrices: Because this subset is not closed under addition, M3(R), it is not a subspace of M3(R).
E. The 3 x 3 matrices that are not invertible Due to the fact that it is not closed under scalar multiplication, this subset is not a subspace of M3(R).
F. The symmetric 3x 3 matrices: This subset, which is closed under addition and scalar multiplication and contains the zero vector, is a subspace of M3(R).
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