If "f" is differentiable and f(1) < f(2), then there is a number "c", in the interval (1, 2) such that f'(c)> 0.
How do we know?Applying the Mean Value Theorem for derivatives, if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the interval (a, b) such that:
f'(c) = (f(b) - f(a)) / (b - a)
In the scenario above, we have that f is differentiable, and that f(1) < f(2).
choosing a = 1 and b = 2.
Then applying the Mean Value Theorem, there exists at least one number c in the interval (1, 2) such that:
f'(c) = (f(2) - f(1)) / (2 - 1)
f'(c) = f(2) - f(1)
We have that f(1) < f(2), we have:
f(2) - f(1) > 0
We can conclude by saying that there exists a number c in the interval (1, 2) such that:
f'(c) = f(2) - f(1) > 0
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Help me please.
Whoever answers right gets brainliest
Distance in the coordinate plane iready
Answer:
Distance in the coordinate plane iready
Step-by-step explanation:
Sure, I can help with distance in the coordinate plane!
The distance between two points (x1, y1) and (x2, y2) in the coordinate plane can be found using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Here's an example:
Let's say we want to find the distance between the points (3, 4) and (6, 8).
We can plug these coordinates into the distance formula:
d = √((6 - 3)^2 + (8 - 4)^2)
Simplifying the expression inside the square root:
d = √(3^2 + 4^2)
d = √(9 + 16)
d = √25
d = 5
Therefore, the distance between the points (3, 4) and (6, 8) is 5 units.
ompute the amount to be paid for each of the four separate invoices assuming that all invoices are paid within the discount period. Terms Payment $ Merchandise (gross) a. $ 8,000 24,500 C. 81,000 17,500 I 2/10, n/60 1/15, EOM 1/10, n/30 3/15, n/45 7,840 20,825 72,900 14,875
Where the above Terms exists, the amount to be paid in each of the above invoices are given as follows;
Invoice 1: $7,840Invoice 2: $24,255Invoice 3: $80,190Invoice 4: $16,975.How would one define the the Terms above?Note that the terms are defined as follows;
Terms A: 2/10, n/60 (2% discount if paid within 10 days, net due in 60 days)
Terms B: 1/15, EOM (1% discount if paid within 15 days, end of month terms)
Terms C: 1/10, n/30 (1% discount if paid within 10 days, net due in 30 days)
Terms D: 3/15, n/45 (3% discount if paid within 15 days, net due in 45 days.
So to compute the amount to be paid for each of the four separate invoices assuming that all invoices are paid within the discount period, we need to calculate the amount of the discount and subtract it from the gross merchandise amount.
For terms 2/10, n/60:
Discount = 2% of $8,000 = $160
Amount to be paid = $8,000 - $160
= $7,840
For terms 1/15, EOM:
Discount = 1% of $24,500 = $245
Amount to be paid = $24,500 - $245
= $24,255
For terms 1/10, n/30:
Discount = 1% of $81,000 = $810
Amount to be paid = $81,000 - $810
= $80,190
For terms 3/15, n/45:
Discount = 3% of $17,500 = $525
Amount to be paid = $17,500 - $525
= $16,975
Therefore, the amount to be paid for each of the four separate invoices assuming that all invoices are paid within the discount period are:
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-3mn(m^2n^3 + 2mn) ASAP PLS SIMPLIFY
Answer:
3mn^4 - 6m^2n^4
2. The directions for sewing a scarf says that you must purchase 1.75 yards of fleece
fabric. The cost per yard is $10.30. How much will you need to spend on fleece fabric?
Answer:
$18.03
Step-by-step explanation:
10.30 x 1.7 = 18.025
round answer to 18.3
Answer: $18.03
Step-by-step explanation:
1.75 x 10.3 = 18.025
18.025 ≈ 18.03
A simple random sample with n = 25 provided a sample mean of 30 and a sample standard deviation of 4. Assume the population is approximately normal. a. Develop a 90% confidence interval for the population mean. b. Develop a 95% confidence interval for the population mean. c. Develop a 99% confidence interval for the population mean. d. What happens to the margin of error and the confidence interval as the confidence level is increased?
Conversely, as the confidence level decreases, the margin of error becomes smaller, and the confidence interval becomes narrower.
What is confidence interval?In statistics, a confidence interval is a range of values that is likely to contain the true value of a population parameter (such as a mean or a proportion), based on a sample from that population. The confidence interval is typically expressed as an interval around a sample statistic, such as a mean or a proportion, and is calculated using a specified level of confidence, typically 90%, 95%, or 99%.
Here,
To develop a confidence interval, we need to use the following formula:
Confidence Interval = sample mean ± margin of error
where the margin of error is calculated as:
Margin of Error = z* (sample standard deviation/ √n)
where z* is the critical value from the standard normal distribution table based on the chosen confidence level.
a. For a 90% confidence interval, the critical value (z*) is 1.645. Thus, the margin of error is:
Margin of Error = 1.645 * (4 / √25) = 1.317
So, the 90% confidence interval for the population mean is:
30 ± 1.317, or (28.683, 31.317)
b. For a 95% confidence interval, the critical value (z*) is 1.96. Thus, the margin of error is:
Margin of Error = 1.96 * (4 / √25) = 1.568
So, the 95% confidence interval for the population mean is:
30 ± 1.568, or (28.432, 31.568)
c. For a 99% confidence interval, the critical value (z*) is 2.576. Thus, the margin of error is:
Margin of Error = 2.576 * (4 / √25) = 2.0656
So, the 99% confidence interval for the population mean is:
30 ± 2.0656, or (27.9344, 32.0656)
d. As the confidence level increases, the margin of error also increases, because we need to be more certain that our interval includes the true population mean. This means that the confidence interval becomes wider as the confidence level increases.
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According to a poll, about % of adults in a country bet on professional sports. Data indicates that % of the adult population in this country is male. Complete parts (a) through (e).
(b) Assuming that betting is independent of gender, compute the probability that an adult from this country selected at random is a male and bets on professional sports.
P(male and bets on professional sports)
0.0568
(c) Using the result in part (b), compute the probability that an adult from this country selected at random is male or bets on professional sports.
P(male or bets on professional sports)
0.5362
(d) The poll data indicated that 7.3% of adults in this country are males and bet on professional sports. What does this indicate about the assumption in part (b)?
A.
The assumption was incorrect and the events are not independent.
Part 5
(e) How will the information in part (d) affect the probability you computed in part (c)? Select the correct choice below and fill in any answer boxes within your choice.
A.
P(males or bets on professional sports) = ?
a) D. No. A person can be both male and bet on professional sports at the same time
How to solveb) If the events A and B are independent, P(A&B) = P(A) x P(B)
P(male and also bets on professional sports) = 0.484x0.13 = 0.0629
c) P(male or bets in professional sports) = P(male) + P(bets in professional sports) - P(male and also bets on professional sports)
= 0.484 + 0.13 - 0.0629
= 0.5511
d) A. The assumption was incorrect and the events are not independent
(if the were independent, the percentage would have been 6.29)
e) A. P(male or bets on professional sports = 0.484 + 0.13 - 0.081
= 0.5330
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The function f(x) is represented by this table of values.
x f(x)
-5 35
-4 24
-3 15
-28
-1
3
0
0
1 -1
Match the average rates of change of fx) to the corresponding intervals.
-8
-7
(-5, -1]
(-4,-1]
[-3, 1]
(2, 1)
HELPPP ASAP
Answer:
-8: (-4, -3]
-7: (-3, -1]
(-5, -1]: (-5, -1]
(-4, -1]: (-4, -1]
[-3, 1]: [-3, 1]
(2, 1): (1, 2]
If anyone could help that would be nice pls :)
Answer:
47 the answer is simply 47
The Venn diagram here shows the cardinality of each set. Use this to find the cardinality of the given set.
n(A)=
The cardinality of set A, n(A) = 29
What is cardinality of a set?The cardinality of a set is the total number of elements in the set
Given the Venn diagram here shows the cardinality of each set. To find the cardinality of set A, n(A), we proceed as follows.
Since the cardinality of a set is the total number of elements in the set, then cardinality of set A , n(A) = 9 + 8 + 3 + 9
= 29
So, n(A) = 29
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Suppose A and B are invertible matrices. Mark each statement as true or false. True means that the statement is true for all invertible matrices A and B.
(In−A)(In+A)=In−A2.
Choose True False
(AB)^−1=A^−1B^−1.
Choose True False
A+B is invertible.
Choose True False
A7 is invertible.
Choose True False
(A+B^)2=A^2+B^2+2AB.
Choose True False
The true statement for all invertible matrices A and B are
1. (In−A)(In+A)=In−A².
2. (AB)⁻¹=A⁻¹B⁻¹
4. A⁷ is invertible.
The given statement is true for all invertible matrices A. To prove this statement, we can expand the left-hand side of the equation as follows:
(In−A)(In+A) = In(In) + In(A) − A(In) − A(A)
= In² + InA − AIn − A²
= In + InA − AIn − A²
= In − A²
Therefore, we have shown that (In−A)(In+A)=In−A2 is true for all invertible matrices A.
The statement is true for all invertible matrices A and B. To prove this statement, we can use the definition of the inverse of a matrix. The inverse of a matrix A is a matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I, where I is the identity matrix. Using this definition, we can show that:
(AB)(A⁻¹B⁻¹) = A(BB⁻¹)A⁻¹ = AIA⁻¹ = AA⁻¹ = I
(B⁻¹A⁻¹)(AB) = B⁻¹(A⁻¹A)B = B⁻¹IB = BB⁻¹ = I
Therefore, we have shown that (AB)⁻¹ = A⁻¹B⁻¹ is true for all invertible matrices A and B.
The statement is false in general. For instance, consider the matrices A = [1 0] and B = [−1 0]. Both A and B are invertible matrices, but A + B = [0 0] which is not invertible as it is not a full rank matrix.
The statement is true for all invertible matrices A. To prove this statement, we can use the fact that the product of invertible matrices is also invertible. Since A is invertible, we can write:
A⁷ = AAAA...A
= A⁶A
= (A⁻¹)⁻¹A⁶A
= (A⁻¹A)⁻¹A⁶A
= IA⁶A
= A⁶
We can repeat this process until we get A⁷ = (A⁻¹)⁻¹. Thus, A⁷ is invertible for all invertible matrices A.
The statement is false in general. To show this, we can use a counterexample. Let A = [1 0] and B = [0 −1]. Then,
(A + B)² = [1 −1][1 −1]
= [0 0]
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500 green hats made in 2 hours how many would be made in 40 hours
Answer:
10,000
Step-by-step explanation:
The first step is to divide 500 hats by 2 hours (500 ÷ 2 = 250)
Second, multiply 250 hats by 40 hours (250 x 40 = 10,000)
Complete the recursive formula of the geometric sequence 10, 6, 3.6, 2.16, ....
a(1) = a(n) = a(n − 1).
The common ratio (r) of this geometric sequence is found by dividing any term by its preceding term, such as:
r = a2/a1 = 6/10 = 0.6
We can use this common ratio to find any term in the sequence using the recursive formula:
a(n) = r * a(n-1)
where a(1) is the first term in the sequence, a(n) is the nth term, and a(n-1) is the (n-1)th term
Using this formula, we can find any term in the sequence. For example:
a(2) = r * a(1) = 0.6 * 10 = 6
a(3) = r * a(2) = 0.6 * 6 = 3.6
a(4) = r * a(3) = 0.6 * 3.6 = 2.16
and so on
Therefore, the complete recursive formula for this geometric sequence is:
a(n) = 0.6 * a(n-1), where a(1) = 10 and a(n) = a(n-1) for all n > 1
Determine the number of elements of a set and represent the number of elements using a symbol
We can represent the number of elements in a set using the symbol |A| or n(A). So, we can say |A| = 5 or n(A) = 5. We can make generalizations about the equality of sets based on the definition above.
What is set?In mathematics, a set is a collection of distinct objects, which can be anything such as numbers, letters, or other mathematical objects. These objects are called the elements or members of the set. Sets can be defined using various methods such as listing the elements, describing properties of the elements, or using set-builder notation. For example, we can define the set of even numbers using set-builder notation as {x | x is an integer and x is divisible by 2}. Sets play a fundamental role in various branches of mathematics, including algebra, geometry, and analysis. They are used to define mathematical structures such as groups, rings, and fields, and to study various mathematical concepts such as functions, relations, and cardinality.
Here,
11.1.4 To determine the number of elements of a set, you need to count the number of distinct elements in the set. For example, if you have a set A = {1, 2, 3, 4, 5}, then the number of elements in set A is 5.
11.1.5 Two sets are equal if they have exactly the same elements. For example, if we have set A = {1, 2, 3} and set B = {3, 2, 1}, then A and B are equal sets because they contain the same elements, even though the order of the elements is different. We can write this as A = B. If two sets are not equal, then we use the symbol ≠ to denote inequality. For example, if set A = {1, 2, 3} and set B = {4, 5, 6}, then A ≠ B.
Some of these generalizations are:
Two sets are equal if and only if they have the same elements.
The order of the elements in a set does not matter for equality.
If two sets have different elements, then they are not equal.
If two sets have the same elements, but with different multiplicities, then they are not equal.
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The scale on a map is 1:320000
What is the actual distance represented by 1cm?
Give your answer in kilometres.
By answering the presented question, we may conclude that Therefore, 1 expressions cm on the map corresponds to a real distance of 3.2 km.
what is expression ?In mathematics, an expression is a collection of integers, variables, and complex mathematical (such as arithmetic, subtraction, multiplication, division, multiplications, and so on) that describes a quantity or value. Phrases can be simple, such as "3 + 4," or complicated, such as They may also contain functions like "sin(x)" or "log(y)". Expressions can be evaluated by swapping the variables with their values and performing the arithmetic operations in the order specified. If x = 2, for example, the formula "3x + 5" equals 3(2) + 5 = 11. Expressions are commonly used in mathematics to describe real-world situations, construct equations, and simplify complicated mathematical topics.
Scale 1:
320000 means that 1 unit on the map represents his 320000 units in the real world.
To find the actual distance represented by 1 cm on the map, you need to convert the units to the same scale.
1 kilometer = 100000 cm
So,
1 unit on the map = 320000 units in the real world
1 cm on the map = (1/100000) km in the real world
Multiplying both sides by 1 cm gives:
1 cm on the map = (1/100000) km * 320000
A simplification of this expression:
1 cm on the map = 3.2 km
Therefore, 1 cm on the map corresponds to a real distance of 3.2 km.
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This problem explores some questions regarding the fishery model
dt
dP
=P(1−P)−h
If you have not yet run the Jupyter notebook please do so now. Find analytical expressions for the two fixed points of the model, in terms of
h
. Give an expression for the stable fixed point. You may assume that the larger fixed point is the stable one. For what values of
h
does there exist a fixed point?
a) The expression for the stable fixed point is P* = (1 + sqrt(1 - 4h)) / 2
b) There exists a fixed point for all values of h less than or equal to 1/4.
The fixed points of the model are the values of P at which dP/dt = 0. Therefore, we need to solve the equation
P(1-P) - h = 0
Expanding the left-hand side, we get
P - P^2 - h = 0
Rearranging, we get a quadratic equation
P^2 - P + h = 0
Using the quadratic formula, the two solutions for P are
P = (1 ± sqrt(1 - 4h)) / 2
a) The larger root is the stable fixed point, as it corresponds to a minimum of the fish population growth function. Therefore, the expression for the stable fixed point is
P* = (1 + sqrt(1 - 4h)) / 2
b) For the model to have a fixed point, the quadratic equation must have real roots. This occurs when the discriminant (the expression inside the square root) is non-negative
1 - 4h ≥ 0
Solving for h, we get
h ≤ 1/4
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The given question is incomplete, the complete question is:
This problem explores some questions regarding the fishery model
dP/dt =P(1−P)−h
If you have not yet run the Jupyter notebook please do so now. Find analytical expressions for the two fixed points of the model, in terms of h
a) Give an expression for the stable fixed point. You may assume that the larger fixed point is the stable one. b) For what values of h does there exist a fixed point?
24 356 ÷ 5 using long division.
Answer:
24 356 ÷ 5 using long division.
Step-by-step explanation:
See the image
a living room rug is 9 ft by 12 ft a strip of floor of equal width is uncovered on all sides of the room if the area od the uncovered floor is 270 ft^2 how wide is the strip
Step-by-step explanation:
To find the width of the strip of uncovered floor, we need to subtract the area of the covered floor from the total area of the room, and then divide by the width of the strip.
The total area of the room is:
9 ft x 12 ft = 108 ft^2
Let's assume the width of the strip is x.
Then the dimensions of the covered floor are:
Length = 9 - 2x Width = 12 - 2x
The area of the covered floor is:
(9 - 2x) x (12 - 2x) = 108 - 30x + 4x^2
We know that the area of the uncovered floor is 270 ft^2, so we can set up the equation:
108 - 30x + 4x^2 = 270
Simplifying and rearranging:
4x^2 - 30x - 162 = 0
Dividing by 2:
2x^2 - 15x - 81 = 0
Using the quadratic formula:
x = [15 ± sqrt(15^2 + 4(2)(81))]/4
x = [15 ± sqrt(1089)]/4
x = [15 ± 33]/4
x = 12 or x = -3/2
Since the width of the strip cannot be negative, we can discard the negative root, and the width of the strip is:
x = 12/2 = 6 ft
Therefore, the strip of uncovered floor is 6 ft wide.
To approximate binomial probability plx > 8) when n is large, identify the appropriate 0.5 adjusted formula for normal approximation. O plx > 7.5) O plx >= 9) O plx > 9) O plx > 8.5)
The appropriate 0.5 adjusted formula for normal approximation is option (d) p(x > 8.5)
The appropriate 0.5 adjusted formula for normal approximation to approximate binomial probabilities when n is large is
P(Z > (x + 0.5 - np) / sqrt(np(1-p)))
where Z is the standard normal variable, x is the number of successes, n is the number of trials, and p is the probability of success in each trial.
To approximate binomial probability p(x > 8) when n is large, we need to use the continuity correction and find the appropriate 0.5 adjusted formula for normal approximation. Here, x = 8, n is large, and p is unknown. We first need to find the value of p.
Assuming a binomial distribution, the mean is np and the variance is np(1-p). Since n is large, we can use the following approximation
np = mean = 8, and
np(1-p) = variance = npq
8q = npq
q = 0.875
p = 1 - q = 0.125
Now, using the continuity correction, we adjust the inequality to p(x > 8) = p(x > 8.5 - 0.5)
P(Z > (8.5 - 0.5 - 8∙0.125) / sqrt(8∙0.125∙0.875))
= P(Z > 0.5 / 0.666)
= P(Z > 0.75)
Therefore, the correct option is (d) p(x > 8.5)
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The given question is incomplete, the complete question is:
To approximate binomial probability p(x > 8) when n is large, identify the appropriate 0.5 adjusted formula for normal approximation. a) p(x > 7.5) b) p(x >= 9) c) p(x > 9) d) p(x > 8.5)
A triangular prism has height 20 cm.
Its triangular face has base 7 cm and height 10 cm.
A. what is the volume of the prism?
B. suppose you triple the height of the prism.what happen to the volume?
C. suppose you triple the base of the triangular face.what happen to the volume?
D. suppose you triple the height of the triangular face.what happen to the volume?
E. suppose you triple all 3 dimensions.what happen to the volume?
Answer:
A. The volume of the triangular prism can be calculated using the formula V = (1/2)bh × h, where b is the base of the triangular face and h is the height of the prism. Thus, V = (1/2)(7 cm)(10 cm) × 20 cm = 700 cubic centimeters.
B. If the height of the prism is tripled to 60 cm, then the new volume would be V' = (1/2)(7 cm)(10 cm) × 60 cm = 2100 cubic centimeters. Thus, the volume is tripled.
C. If the base of the triangular face is tripled to 21 cm, then the new volume would be V' = (1/2)(21 cm)(10 cm) × 20 cm = 2100 cubic centimeters. Thus, the volume is tripled.
D. If the height of the triangular face is tripled to 30 cm, then the new volume would be V' = (1/2)(7 cm)(30 cm) × 20 cm = 2100 cubic centimeters. Thus, the volume is tripled.
E. If all three dimensions (base, height of triangular face, and height of prism) are tripled, then the new volume would be V' = (1/2)(21 cm)(30 cm) × 60 cm = 18900 cubic centimeters. Thus, the volume is multiplied by a factor of 27.
Please see attached question
Using graphs, we can see that the point (4,2) can be a coordinate where y will represent x.
What are graphs?The graph is simply a structured representation of the data. The numerical information gathered through observation is referred to as data.
If there is just one value of y (output) for every value of x, the relationship between x and y is said to be a function (input).
In other words, there can only be one value of y for each value of x.
Determine each plotted point's coordinates first:
(-4,4)
(-2,3)
(0,1)
(2, -1)
(3,0)
The following point cannot have any of the x-coordinates of the displayed points, which are -4, -2, 0, 2, and 3.
Options include:
A (0,1) →The relationship cannot be regarded as a function at this stage as the x-coordinate zero already has a corresponding value of y.
B (2,2) →Although there is already a value of y for the location x=2, the relationship cannot be regarded as a function at this point.
C (3,4) →Although there is already a value of y for the location x=3, the relationship cannot be regarded as a function at this point.
D (4,2) → The relationship will still be regarded as a function even though there are no points on the graph with the coordinates x=4 displayed.
Therefore, option D (4,2) is the point where y will represent x.
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The complete question:
Please see attached question
In the following alphanumeric series, what letter comes next? V, Q, M, J, H, …
According to the given information, the letter that comes next in the given alphanumeric series is "N".
What is alphanumeric series?
An alphanumeric series is a sequence of letters and/or numbers that follows a certain pattern or rule. For example, "A, B, C, D, E..." is an example of an alphabetical series, and "1, 3, 5, 7, 9..." is an example of a numerical series. An alphanumeric series may combine both letters and numbers, such as "A1, B2, C3, D4, E5...". The pattern or rule followed by an alphanumeric series may be based on numerical or alphabetical order.
The given series V, Q, M, J, H, ... follows a pattern where each letter is the 6th letter from the previous letter. So, the next letter in the series would be 6 letters after H, which is N.
Therefore, the letter that comes next in the given alphanumeric series is "N".
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FILL IN THE BLANK. The population parameters that describe the y-intercept and slope of the line relating y and x, respectively, are _____. Group of answer choices B0 and B1 y and x a and b a and B
The equation for a linear regression model is represented as: y = B0 + B1x + error.
What is Slope ?
Slope refers to the measure of steepness of a line. It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on the line. It is also called the gradient of the line.
In statistics, a linear regression model is used to describe the relationship between a dependent variable (usually denoted as "y") and one or more independent variables (usually denoted as "x").
The equation for a linear regression model is represented as: y = B0 + B1x + error
where B0 is the y-intercept, B1 is the slope of the line relating y and x, and error is the random error term. The population parameters B0 and B1 are the true values of the y-intercept and slope that would be obtained if the entire population were studied, rather than just a sample.
These parameters are estimated using the sample data, and the resulting estimates are used to construct the regression equation. The estimates of the population parameters are based on the assumption that the sample is representative of the population, and that the errors are normally distributed with constant variance.
Therefore, The equation for a linear regression model is represented as: y = B0 + B1x + error
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calculate the are of given figure
Whenever we construct a confidence interval for the population mean, the margin of error includes the standard error of x bar and theA. sampling biasB. nonresponse biasC. z or t value associated with a 95% confidence levelD. desired level of confidence
The margin of error in a confidence interval for the population mean includes the standard error of the sample mean and the z or t value associated with a certain level of confidence, usually 95% or 99%. Therefore, the correct answer is (C).
Sampling bias and nonresponse bias are potential sources of error in survey or study design and data collection, but they are not directly related to the construction of a confidence interval.
The desired level of confidence is a key input for determining the z or t value used in the calculation of the margin of error, but it is not included in the margin of error itself. The correct answer is z or t value associated with a 95% confidence level.
So, the correct option is (C).
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6x+16=8x-18 i need x
Answer:
x = 17
Step-by-step explanation:
Subtract 6x from both sides:
2x - 18 = 16
Add 18 on both sides to isolate the variable:
2x = 18 + 16
2x = 34
Divide by 2: x = 17
The standard deviation of the scores on a skill evaluation test is 497
points with a mean of 1754
points.
If 302 tests are sampled, what is the probability that the mean of the sample would differ from the population mean by less than 44
points? Round your answer to four decimal places.
Answer:
497/√302 = 49.7
z-score = (44-0)/49.7 = 0.88
Probability = 0.8133
LOOK AT THE PHOTO PLS
Answer:
0.54545454545
Step-by-step explanation:
Find the maximum/minimum value of the quadratic function q² + 22q = y - 85 by
completing the square method.
O-36
O-48
C
-24
-64
Step-by-step explanation:
To find the maximum/minimum value of the quadratic function q² + 22q = y - 85, we can complete the square as follows:
q² + 22q = y - 85
q² + 22q + 121 = y - 85 + 121 (adding (22/2)² = 121 to both sides)
(q + 11)² = y + 36
Now, we have a square of a binomial on the left side, which means the minimum value of the quadratic function is y + 36, and it occurs when (q + 11) = 0. Thus, the minimum value is:
y + 36 = 36 - 85 = -49
Similarly, the maximum value of the quadratic function occurs when (q + 11) = 0, but this time the value of y will be as large as possible. The largest possible value of y occurs when STU is a permutation of the digits 9, 8, and 7 (since PQR must be 4, 0, and 3 in some order). Thus, the maximum value is:
y + 36 = 9 + 8 + 7 - 85 + 36 = -25
Therefore, the options are incorrect and the correct answers are:
Minimum value: -49
Maximum value: -25
What is the length of the distance between the two points of (6,-2)
and (3, 4)?
O √13
O √45
O √65
O √117
Answer:
[tex]\sqrt {45}[/tex]
Step-by-step explanation:
Distance between two points, (x1, y1) and (x2, y2) in a 2D cartesian coordinate is given by
[tex]d = \sqrt {(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}[/tex]
here the points are (6, - 2) and (3, 4)
We get
[tex]d = \sqrt {(3 - 6)^2 + (4 - (-2))^2}\\\\d = \sqrt {(-3)^2 + (6)^2}\\\\d = \sqrt {{9} + {36}}\\\\d = \sqrt {45}[/tex]