a(a - b)+b(a - b) + (a - b)²​

The probability that a student chosen randomly from the class has a brother is approximately 0.143 or 14.3%.

To find the probability that a student chosen randomly from the class has a brother, we need to look at the number of students who have a brother and divide it by the total number of students in the class.

From the given data table, we see that there are a total of 4+2+12+10=28 students in the class. Out of these, 4 students have a brother. Therefore, the probability that a student chosen randomly from the class has a brother is:

P(having a brother) = Number of students having a brother / Total number of students

= 4 / 28

= 1/7

≈ 0.143

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For the line segment, the measure of angle BOD is 90°.

We will draw a circle passing through points A, B, C, and D. Since AC is parallel to BD, this circle will be the circumscribed circle of quadrilateral ABCD.

Now, let's consider the angles formed by the intersection of the circle and the lines AB and CD. We know that angle CAB is equal to half the arc AC of the circle, and angle CDB is equal to half the arc BD.

Since AC is parallel to BD, arc AC is congruent to arc BD. Therefore, angle CAB is equal to angle CDB.

Using this information, we can find the measure of angle AOB, which is equal to angle CAB + angle CDB. Substituting the given values, we get angle AOB = 35° + 55° = 90°.

Finally, we can use the fact that angle AOB and angle COD are supplementary angles (they add up to 180°) to find the measure of angle BOD.

Angle BOD = 180° - angle AOB

Substituting the value of angle AOB, we get

Angle BOD = 180° - 90° = 90°

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Complete Question:

Line segments AB and CD intersect at O such that AC∣∣DB. If ∠CAB=35° and ∠CDB=55°, find ∠BOD.

The distance from point A to point B is approximately 998 feet (rounded to the nearest foot).

Let's denote the distance from point A to the lighthouse as "x", and the distance from point B to the lighthouse as "y". Also, let's denote the height of the lighthouse as "h". Then we have the following diagram:

Lighthouse

    |\

    | \

    |  \ h

    |   \

    |θ2 \

    |____\

      x   y

      A   B

From the diagram, we can see that:

tan(6°) = h/x (equation 1)

and

tan(4°) = h/y (equation 2)

We need to find the value of "d", the distance from point A to point B. We can use the following equation:

d^2 = x^2 + y^2 (equation 3)

We can solve equation 1 for h:

h = x tan(6°)

Substitute this into equation 2:

x tan(6°) / y = tan(4°)

Solve for y:

y = x tan(6°) / tan(4°)

Substitute this into equation 3:

d^2 = x^2 + (x tan(6°) / tan(4°))^2

Simplify:

d^2 = x^2 (1 + tan^2(6°) / tan^2(4°))

Solve for d:

d = x sqrt(1 + tan^2(6°) / tan^2(4°))

Substitute the given values:

d = 996 sqrt(1 + tan^2(6°) / tan^2(4°))

Using a calculator, we get:

tan(6°) / tan(4°) = 0.1051

So,

d = 996 sqrt(1 + 0.1051^2) ≈ 998.38 feet

Therefore, the distance from point A to point B is approximately 998 feet (rounded to the nearest foot).

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The mathematical phrase 5 + 2 × 18 is an example of an arithmetic expression.

To solve this expression, follow the order of operations (PEMDAS/BODMAS):

1. Parentheses/Brackets (P/B)
2. Exponents/Orders (E/O)
3. Multiplication and Division (M/D)
4. Addition and Subtraction (A/S)

Your expression: 5 + 2 × 18

Step 1: No parentheses/brackets to solve.

Step 2: No exponents/orders to solve.

Step 3: Solve multiplication: 2 × 18 = 36

Step 4: Solve addition: 5 + 36 = 41

So, the value of the expression 5 + 2 × 18 is 41.

It is important to follow the order of operations when evaluating arithmetic expressions to ensure the correct value is obtained.

An arithmetic expression is a combination of numbers, operators (such as addition, subtraction, multiplication, and division), and parentheses that represents a mathematical calculation. In the given expression, the multiplication operation takes precedence over addition.

According to the order of operations (PEMDAS/BODMAS), multiplication is performed before addition. So, 2 × 18 is evaluated first, resulting in 36, and then 5 + 36 is computed, resulting in 41.

Therefore, the value of the expression is 41. Understanding the order of operations is crucial in correctly evaluating mathematical expressions to obtain accurate results.

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The probability that Anne could pass both mathematics and English courses is 0.19 or 19%.

To find the probability that Anne could pass both mathematics and English, we can use the formula for the probability of the union of two events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B), where A is the event of passing mathematics, B is the event of passing English, and A ∩ B is the event of passing both courses.
We are given:
P(A) = probability of passing mathematics = 0.42
P(B) = probability of passing English = 0.47
P(A ∪ B) = probability of passing at least one course = 0.7

Now we need to find the probability of passing both courses, P(A ∩ B).
Using the formula, we have:
0.7 = 0.42 + 0.47 - P(A ∩ B)
To find P(A ∩ B), we rearrange the equation:
P(A ∩ B) = 0.42 + 0.47 - 0.7
Now, calculate the probability:
P(A ∩ B) = 0.19
So, the probability that Anne is 0.19 or 19%.

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The largest possible area of a square that can be made from any of the three pieces is [tex](400 cm)^{2}[/tex]

To find the largest possible area of a square that can be made from any of the three similar bars of length 200 cm, 300 cm, and 360 cm, you need to first determine the greatest common divisor (GCD) of their lengths.

Step 1: Find the GCD of 200, 300, and 360.
The prime factorization of 200 is [tex](2^{3})(5^{2})[/tex], of 300 is [tex](2^{2})(3)(5^{2})[/tex], and of 360 is [tex](2^{3})(3^{2})(5)[/tex]. The GCD is the product of the lowest powers of common factors, which is [tex](2^{2})5=20[/tex].

Step 2: Determine the side length of the largest square.
Since the bars are cut into equal pieces with a length of 20 cm (the GCD), the largest square will have a side length of 20 cm.

Step 3: Calculate the largest possible area of the square.
The area of the square can be found by multiplying the side length by itself: [tex]Area = (side)^{2}[/tex].
[tex]Area = (20 cm)(20 cm) = (400 cm)^{2}[/tex].

So, the largest possible area of a square that can be made from any of the three pieces is [tex](400 cm)^{2}[/tex].

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The probability of having at least one failure in five trials is approximately 0.83193 or 83.193%.

To calculate the probability of at least one failure, we first need to find the probability of having zero failures in five trials, which is equal to (0.3)^5 or 0.00243. Then, we subtract this value from 1 to obtain the probability of having at least one failure. This is because the sum of the probabilities of all possible outcomes should be equal to 1.

In this case, we can see that the probability of having at least one failure in five trials is quite high, at approximately 83%. This means that it is more likely than not that there will be at least one failure in a series of five trials with a success rate of 30%.

The probability of having at least one failure in five trials of a binomial experiment with a success rate of 30% can be calculated as follows:

1 - (probability of having zero failures in five trials)

= 1 - (0.7)^5

= 1 - 0.16807

= 0.83193

Therefore, the probability of having at least one failure in five trials is approximately 0.83193 or 83.193%.

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Katy and Colleen each choose a number from 3, 6, or 8. If the sum is even, Katy pays Colleen the sum in dimes, and if odd, Colleen pays Katy the sum in dimes. There are 9 possible outcomes with payments ranging from 3 to 16 dimes.

If Katy writes down 3, then Colleen has two choices, either write down 3 to make the sum even or 6 to make it odd. If Colleen writes down 3, the sum is even, and Katy pays Colleen 6 dimes. If Colleen writes down 6, the sum is odd, and Colleen pays Katy 3 dimes.

If Katy writes down 6, then Colleen has two choices, either write down 3 to make the sum odd or 8 to make it even. If Colleen writes down 3, the sum is odd, and Colleen pays Katy 6 dimes. If Colleen writes down 8, the sum is even, and Katy pays Colleen 14 dimes.

If Katy writes down 8, then Colleen has two choices, either write down 3 to make the sum odd or 6 to make it even. If Colleen writes down 3, the sum is odd, and Colleen pays Katy 8 dimes. If Colleen writes down 6, the sum is even, and Katy pays Colleen 14 dimes.

Therefore, the possible outcomes and their corresponding payments are

3 + 3: odd, Colleen pays Katy 3 dimes

3 + 6: even, Katy pays Colleen 6 dimes

3 + 8: odd, Colleen pays Katy 8 dimes

6 + 3: odd, Colleen pays Katy 6 dimes

6 + 6: even, Katy pays Colleen 14 dimes

6 + 8: even, Katy pays Colleen 14 dimes

8 + 3: odd, Colleen pays Katy 8 dimes

8 + 6: even, Katy pays Colleen 14 dimes

8 + 8: even, Katy pays Colleen 16 dimes.

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Answer:

pls mrk me brainliest (⁠´⁠(⁠ｪ⁠)⁠｀⁠）

Looking at the plot, we can see that the stems range from 60 to 89, with each stem representing a group of ten pounds. The leaves represent the remaining single digits, indicating the exact weight of each tuna. There are 4 tuna that weigh less than 90 pounds

Based on the stem-and-leaf plot of the weights of yellowfin tuna caught during a fishing contest, we can count the number of tuna that weigh less than 90 pounds.

To determine the number of tuna that weigh less than 90 pounds, we need to look at the stems that are less than 9. This includes stems 6, 7, and 8. The leaves associated with these stems show the weights of the tuna that are less than 90 pounds. We can count the number of leaves associated with these stems to determine the number of tuna that weigh less than 90 pounds.

In this case, there are 4 tuna that weigh less than 90 pounds. Two of them weigh 88 pounds and the other two weigh 87 pounds. Therefore, we can conclude that there are 4 tuna that weigh less than 90 pounds in the fishing contest based on the stem-and-leaf plot.

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The sum of the first 8 terms of the sequence is 5100. Rounded to the nearest hundredth, this is 5100.00

To find the sum of the first 8 terms of the sequence 20, 40, 80,..., we need to use the geometric series formula:

S = a(1 - r^{n}) / (1 - r)

where S is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

In this case, a = 20 (the first term), r = 2 (the common ratio, since each term is twice the previous one), and n = 8 (since we want to find the sum of the first 8 terms).

So plugging these values into the formula, we get:

S = 20(1 - 2^8) / (1 - 2)
S = 20(1 - 256) / (-1)
S = 20(255)
S = 5100

Therefore, the sum of the first 8 terms of the sequence is 5100. Rounded to the nearest hundredth, this is 5100.00.

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The area of the surface is [tex]\sqrt{\frac{15}{4}}\times \pi[/tex]  unit square.

To find the area of the surface, we need to first find the intersection curve between the plane and the cylinder.
From the equation of the cylinder, we know that [tex]x^2 + y^2[/tex] = 9200.

We can substitute [tex]x^2 + y^2[/tex] for [tex]r^2[/tex] and rewrite the equation as [tex]r^2[/tex] = 9200.
Next, we can rewrite the equation of the plane as

z = 12 - 4x - 3y.
Now, we can substitute 12 - 4x - 3y for z in the equation [tex]r^2[/tex] = 9200, giving us:
[tex]x^2 + y^2[/tex] = 9200 - [tex](12 - 4x - 3y)^2[/tex]
Expanding and simplifying, we get:
[tex]x^2 + y^2[/tex] = [tex]16x^2 + 24xy + 9y^2 - 24x - 36y + 884[/tex]
Simplifying further, we get:
[tex]15x^2 + 24xy + 8y^2 - 24x - 36y + 884 = 0[/tex]
We can recognize this as the equation of an ellipse:

To find the area of the surface, we need to find the area of this ellipse that lies within the cylinder.
To do this, we can first find the major and minor axes of the ellipse.

We can rewrite the equation as:
[tex]15(x - \frac{4}{5})^2[/tex] + 8([tex]y[/tex] - [tex]\frac{9}{10}[/tex][tex])^{2}[/tex] = 1

So the major axis has length [tex]2/\sqrt{15}[/tex] unit and the minor axis has length [tex]\frac{2}{\sqrt{8} }[/tex] unit.
The area of the ellipse is then given by:
A = π x ([tex]\frac{1}{2}[/tex] x [tex]\frac{2}{\sqrt{15} }[/tex] x ([tex]\frac{1}{2}[/tex] x [tex]\frac{8}{\sqrt{8} }[/tex])
Simplifying we get:
A = π x ([tex]\sqrt{\frac{2}{15} }[/tex]) x ([tex]\sqrt{\frac{2}{8} }[/tex])
A = π x ([tex]\sqrt{\frac{1}{60} }[/tex])
A = [tex]\sqrt{\frac{15}{4}} \times \pi[/tex] unit square

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To find the rate of change of the volume of the cone, we need to use the formula for the volume of a cone:

V = (1/3)πr^2h

Taking the derivative of both sides with respect to time, we get:

dV/dt = (1/3)π[2rh(dr/dt) + r^2(dh/dt)]

Substituting the given values:

r = 40 in (radius is increasing at a rate of 4 in/s)
h = 20 in (height is decreasing at a rate of 3 in/s)
dr/dt = 4 in/s
dh/dt = -3 in/s

Plugging these into the formula:

dV/dt = (1/3)π[2(40)(20)(4) + (40)^2(-3)]

dV/dt = (1/3)π[3200 - 4800]

dV/dt = (1/3)π(-1600)

dV/dt = -1681.99 in^3/s

Therefore, the volume of the cone is decreasing at a rate of approximately 1681.99 cubic inches per second when the radius is 40 inches and the height is 20 inches.
To find the rate at which the volume of the cone is changing, we can use the formula for the volume of a cone (V = (1/3)πr^2h) and differentiate it with respect to time (t).

Given:
dr/dt = 4 inches per second (increasing radius)
dh/dt = -3 inches per second (decreasing height)
r = 40 inches
h = 20 inches

First, let's differentiate the volume formula with respect to time:
dV/dt = d/dt[(1/3)πr^2h]

Using the product and chain rules, we get:
dV/dt = (1/3)π(2r(dr/dt)h + r^2(dh/dt))

Now, plug in the given values:
dV/dt = (1/3)π(2(40)(4)(20) + (40)^2(-3))

Simplify:
dV/dt = (1/3)π(6400 - 4800)

dV/dt = (1/3)π(1600)

Finally, calculate the rate:
dV/dt ≈ 1675.52 cubic inches per second

So, the volume of the cone is changing at a rate of approximately 1675.52 cubic inches per second.

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If each of the 3 friends orders a veggie mushroom cheddar burger for 11%, the cost of each burger would be:

11% of $1.00 = $0.11

Since the prices are listed in fractions of a dollar, we can express the cost of each burger as 11/100 of a dollar.

So, the total cost of 3 veggie mushroom cheddar burgers would be:

3 x 11/100 = 33/100 = $0.33

Assuming that the glass of water is free, the total bill before taxes would be $0.33 for the 3 burgers. However, it's important to note that this calculation is based on the assumption that the prices are listed in fractions of a dollar, which may not be the case. If the prices are listed in a different unit, the calculation would need to be adjusted accordingly.

Based on the given information, Yasmina should use Sally's Saddles if she expects to give 200 rides and wants to make the most profit.

To determine which company Yasmina should use to offer horseback rides at her school's Spring Fair, we need to compare the costs and revenues associated with each option.

First, let's consider Polly's Ponies. They charge $100 for a small pony and Yasmina can charge children $2 for a ride. To break even with this option, Yasmina would need to give 50 rides ($100 / $2 per ride). If she expects to give 200 rides, she would earn $400 in revenue ($2 per ride x 200 rides) and have a profit of $300 ($400 revenue - $100 rental fee).

Next, let's consider Sally's Saddles. They charge $240 for a larger horse and Yasmina can charge children $3 for a ride. To break even with this option, Yasmina would need to give 80 rides ($240 / $3 per ride). If she expects to give 200 rides, she would earn $600 in revenue ($3 per ride x 200 rides) and have a profit of $360 ($600 revenue - $240 rental fee).

Therefore, if Yasmina wants to make the same amount of profit regardless of which company she uses, she would need children to take a total of 125 rides ((($240 rental fee for Sally's Saddles - $100 rental fee for Polly's Ponies) / ($3 per ride - $2 per ride)). If she expects to give 200 rides, she should use Sally's Saddles since she will make a higher profit of $360 compared to $300 with Polly's Ponies.

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Answer:

k = -3.

Step-by-step explanation:

To answer this question, we need to use our own knowledge and information. Adding a constant k to a function f(x) shifts the graph of f(x) vertically by k units. If k is positive, the graph moves up. If k is negative, the graph moves down. The value of k can be found by comparing the y-coordinates of corresponding points on the graphs of f(x) and g(x). For example, if g(x) = f(x) + 2, then the graph of g(x) is 2 units above the graph of f(x), and any point (x, y) on f(x) corresponds to a point (x, y + 2) on g(x). Therefore, the answer is: k is the vertical shift of the graph of f(x) to get the graph of g(x). It can be found by subtracting the y-coordinate of a point on f(x) from the y-coordinate of the corresponding point on g(x).

Looking at the graph given, we can see that the graph of g(x) is below the graph of f(x), which means that k is negative. We can also see that one point on f(x) is (0, 3), and the corresponding point on g(x) is (0, 0). Using the formula above, we get:

k = y_g - y_f

k = 0 - 3

k = -3

Therefore, the correct option is k = -3.

The probability of the the sample proportion of surgeons will be given as 1.

The z-score is a dimensionless variable that is used to express the signed, fractional number of standard deviations by which an event is above the mean value being measured. It is also known as the standard score, z-value, and normal score, among other terms. Z-scores are positive for values above the mean and negative for those below the mean.

For this case we can define the population proportion p as "true proportion of surgeons" and we can check if we can use the normal approximation for the distribution of p,

1) np = 738 x  0.44 = 324.72 > 10

2) n(1 - p) = 738 x  (1 - 0.44) = 413.28 > 10

3) Random sample: We assume that the data comes from a random sample Since we can use the normal approximation the distribution for P is given by:

psimN(p,[tex]\sqrt{\frac{p(1-p)}{n} }[/tex])

With the following parameters:

Hp = 0.44

[tex]\sigma_p=\sqrt{\frac{0.44(1-0.44)}{738} }[/tex]

= 0.01827

And we want to find this probability:

P(p > 0.04)

And we can use the z score formula given by:

[tex]z=\frac{p-\mu}{\sigma}[/tex]

And if we calculate the z score for p = 0.39 we got:

[tex]z=\frac{0.04-0.44}{0.01827}[/tex] = -21.893

And we can find this probability using the complement rule and the normal standard table or excel and we got:

P(p > 0.04) = P(Z > -21.893) = 1 − P(Z < −21.893) = 1 - 0 = 1.

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The conjecture "X, Y, Z and W are noncolinear" is always true when given that line segments XY and ZW intersect at point A. So option D is the correct answer.

When line segments XY and ZW intersect at point A, it means that X, Y, Z, and W do not all lie on the same line. Since they do not all lie on the same line, they are considered non-collinear.

So the correct answer is option D. X, Y, Z and W are noncolinear. ​

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The number of minutes you can use the card is 9 minutes

From the question, we have the following parameters that can be used in our computation:

Using the above as a guide, we have the following:

f(x) = 0.25x + 0.75

If the total cost of the card is $5.00, the number of minutes you is

0.25x + 0.75 = 5

So, we have

0.25x = 4.25

Divide by 0.25

x = 9

Hence, the number of minutes is 9

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The best group of people for Tom to survey would be homeowners in his town, as they are more likely to have a pool in their backyard.

To determine the percentage of people in his town that own a pool, Tom should survey a random sample of residents within the town. This will help him gather accurate and representative data about pool ownership in the area for his potential pool cleaning business.

Tom can also narrow down his survey to neighborhoods that are known to have a higher concentration of pool owners. This will give him a more accurate percentage of pool owners in his town and help him make an informed decision about opening a pool cleaning business.

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The interval 4:00-4:59 represents the most number of cars.

How to determine the interval with the most number of cars based on the given time ranges?

To determine the interval that represents the most number of cars, we need to analyze the given options and find the one with the highest number of cars.

Unfortunately, we don't have any data about the actual number of cars during those intervals. Therefore, we cannot provide a definitive answer to this question. We could only make an educated guess based on certain assumptions.

For instance, if we assume that the traffic is usually higher during rush hour, we could say that the intervals between 4:00-4:59 and 3:00-3:59 are more likely to have the highest number of cars. However, without additional information or data, we cannot provide a more accurate answer.

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The angle measure is given as follows:

θ = 1/3 radians.

The circumference of a circle of radius r is given by the equation presented as follows:

C = 2πr.

The radius is half the diameter, hence it is given as follows:

r = 27 ft. (half the diameter).

Hence the circumference is given as follows:

C = 54π cm.

The fraction represented by a distance of 9 ft is given as follows:

9/54π = 1/6π

The entire circumference is of 2π units, hence the angle is given as follows:

1/(6π) x 2π = 1/3 radians.

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The required answer is a possible scenario where BC (c) is less than AC * sin(angle A)

To justify if it is possible to have BC < AC sin(angle A) for an acute triangle ABC, let's consider the sine formula for a triangle.

The sine formula for a triangle states that:

a/sin(A) = b/sin(B) = c/sin(C)

where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the angles opposite to those sides, respectively.

Now let's isolate side BC (b) in the equation:

b = c * sin(B) / sin(C)

Since triangle ABC is acute, all angles A, B, and C are less than 90°. Therefore, sin(B) and sin(C) will be positive values between 0 and 1.

Let's now compare BC (b) to ACsin(angle A):

b < AC * sin(A)

c * sin(B) / sin(C) < AC * sin(A)

We can rewrite the inequality in terms of angle C:

sin(B) / sin(C) < (AC * sin(A)) / c

Now let's recall that angle C is the angle opposite to side AC (c), and angle B is the angle opposite to side BC (b). Since sine is a positive increasing function for acute angles (0° to 90°), it follows that the sine of a larger angle will result in a larger value.

As angle C is opposite to the longer side (AC), angle C > angle B. Therefore, sin(C) > sin(B), and their reciprocals will have the opposite relationship:

1 / sin(C) < 1 / sin(B)

Now, let's multiply both sides of the inequality by c * sin(B):

c < AC * sin(A)

This inequality represents a possible scenario where BC (c) is less than AC * sin(angle A), justifying the initial claim. So, yes, it is possible to have BC < AC sin(angle A) for an acute triangle ABC.

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Answer:

[tex]y=log_{9} (x)[/tex]   (the middle choice)

Step-by-step explanation:

Key Concepts

Concept 1. Exponential vs logarithm

Concept 2. Logarithm rules

Concept 1. Exponential vs logarithm

The first two choices are logarithmic functions whereas the last function is an exponential function.  The graph cannot be that of an exponential function because exponential functions cannot cross the x-axis (an asymptote) unless a shift transformation is applied (which would look like adding or subtracting a constant number at the end of the equation.

A second way to verify is to simply input 2 into the function.  The number 2 raised to the 9 power is 2*2*2*2*2*2*2*2*2=512, but the graph clearly does not have a height of 512 when the input is 2.  Therefore, the correct answer cannot be the last choice.

Concept 2. Logarithm rules

One important rule for logarithms is that a number input into logarithm that matches the base of the logarithm will yield 1 as a result.  In other words:

For all real numbers b, such that b is positive and not equal to 1, [tex]log_{b}(b)=1[/tex]

Observe that for the first option, this means that [tex]log_{\frac{1}{9}}(\frac{1}{9})=1[/tex].  However, for an input of 1/9, the output is still below the x-axis -- a negative output -- clearly not 1.

Observe that for the second option, this means that [tex]log_{9}(9)=1[/tex], and that for an input of 9, the output on the graph is at a height of 1.

Therefore, the correct function for this question must be the middle option.

Part a): To find f(10)(a), we need to take the 10th derivative of the Taylor series of f(x) at x=a. Since the Taylor series is given by ¿ (-1)n (x-a)^(2n), we need to differentiate this series 10 times with respect to x. Each differentiation will give us a factor of (2n) or (2n-1) times the previous term, and the (-1)n factor will alternate between positive and negative values.

Starting with n=0, we get:

f(x) = ¿ (-1)^n (x-a)^(2n)
f'(x) = ¿ (-1)^n (2n)(x-a)^(2n-1)
f''(x) = ¿ (-1)^n (2n)(2n-1)(x-a)^(2n-2)
f'''(x) = ¿ (-1)^n (2n)(2n-1)(2n-2)(x-a)^(2n-3)
...

After 10 differentiations, we end up with:

f^(10)(x) = ¿ (-1)^n (2n)(2n-1)(2n-2)...(2n-8)(2n-9)(x-a)^(2n-10)

To evaluate this at x=a, we can replace all instances of (x-a) with 0, and we end up with:

f^(10)(a) = ¿ (-1)^n (2n)(2n-1)(2n-2)...(2n-8)(2n-9)(a-a)^(2n-10)
f^(10)(a) = ¿ (-1)^n (2n)(2n-1)(2n-2)...(2n-8)(2n-9)(0)
f^(10)(a) = 0

Therefore, f(10)(a) = 0.

Part b): To find f(11)(a), we need to differentiate the series from part a one more time. We start with the series:

f(x) = ¿ (-1)^n (x-a)^(2n)

and differentiate it 11 times:

f(x) = ¿ (-1)^n (x-a)^(2n)
f'(x) = ¿ (-1)^n (2n)(x-a)^(2n-1)
f''(x) = ¿ (-1)^n (2n)(2n-1)(x-a)^(2n-2)
f'''(x) = ¿ (-1)^n (2n)(2n-1)(2n-2)(x-a)^(2n-3)
...
f^(10)(x) = ¿ (-1)^n (2n)(2n-1)(2n-2)...(2n-8)(2n-9)(x-a)^(2n-10)

and then differentiate once more:

f^(11)(x) = ¿ (-1)^n (2n)(2n-1)(2n-2)...(2n-8)(2n-9)(2n-10)(x-a)^(2n-11)

To evaluate this at x=a, we can replace all instances of (x-a) with 0, and we end up with:

f^(11)(a) = ¿ (-1)^n (2n)(2n-1)(2n-2)...(2n-8)(2n-9)(2n-10)(a-a)^(2n-11)
f^(11)(a) = ¿ (-1)^n (2n)(2n-1)(2n-2)...(2n-8)(2n-9)(2n-10)(0)
f^(11)(a) = 0

Therefore, f(11)(a) = 0.

Given the Taylor series of function f(x):

f(x) = Σ(-1)^n * (x-a)^(2n) / (n+2), where the summation runs from n = 0 to infinity and is centered at x = a.

Part a) To find f(10)(a), we need to determine the 10th derivative of f(x) with respect to x, evaluated at x = a.

Notice that only even terms contribute to the derivatives. The 10th derivative of the Taylor series will have n = 5 (since 2*5 = 10):

f(10)(a) = (-1)^5 * (a-a)^(2*5) / (5+2) = (-1)^5 * 0^10 / 7 = 0

Part b) To find f(11)(a), we need to determine the 11th derivative of f(x) with respect to x, evaluated at x = a. However, the given Taylor series only contains even powers of (x-a), and taking odd derivatives will result in terms with odd powers. Therefore, all odd derivatives, including the 11th derivative, will be 0:

f(11)(a) = 0

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Using the identity: sin(A + B) = sin A cos B + cos A sin B, we can rewrite the expression as follows:

Sin 7A * Cos 3A = (sin 4A + sin 10A)/2 * (cos 2A + cos A)/2

Expanding this expression using the same identity, we get:

= (sin 4A * cos 2A + sin 4A * cos A + sin 10A * cos 2A + sin 10A * cos A)/4

Now, using the identity sin 2A = 2 sin A cos A, we can simplify further:

= (1/2) sin 2A * cos 2A + (1/2) sin 2A * cos 8A + (1/2) sin 6A * cos 2A + (1/2) sin 6A * cos 8A

Therefore, Sin 7A * Cos 3A can be written as:

(1/2) sin 2A * cos 2A + (1/2) sin 2A * cos 8A + (1/2) sin 6A * cos 2A + (1/2) sin 6A * cos 8A

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The problem is asking for the probability of construction given that Max had to stop for a passing train on his route to work. This can be solved using Bayes' theorem, which states that the probability of A given B is equal to the probability of B given A multiplied by the probability of A, divided by the probability of B.

In this case, let A be the event that there is construction on Max's route, and let B be the event that Max has to stop for a passing train. We are looking for the probability of A given B.

Using Bayes' theorem, we have:

P(A|B) = P(B|A) * P(A) / P(B)

We know that P(B) = 0.5, the probability that Max has to stop for a passing train. We also know that P(B|A) = 0.4, the probability that there is construction and Max has to stop for a passing train.

To find P(A), the probability of construction on Max's route, we need to use the complement of the event A, which is the probability that there is no construction:

P(not A) = 1 - P(A) = 1 - 0.4 = 0.6

Finally, we can plug in the values and solve for P(A|B):

P(A|B) = 0.4 * 0.4 / 0.5 = 0.32

Therefore, the probability that there was construction if Max had to stop for a passing train on his route to work is 0.32 or 32%.

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The inverse of the function y= (2/3)x^5-10 is y = [3/2(x + 10)]^1/5

From the question, we have the following parameters that can be used in our computation:

y= (2/3)x^5-10

Swap the ocurrence of x and y

so, we have the following representation

x = (2/3)y^5-10

Next, we have

(2/3)y^5 = x + 10

This gives

y^5 = 3/2(x + 10)

Take the fifth root of both sides

y = [3/2(x + 10)]^1/5

Hence, the inverse function is y = [3/2(x + 10)]^1/5

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The minimum value of the function is -5 and the maximum value of the function is 15.

The minimum and maximum value of the given trigonometric function f(x)=10sin(2/5x)+5 can be determined by analyzing the amplitude and period of the sine function. The amplitude of the sine function is 10, which means that the maximum value of the function is 10+5=15 and the minimum value is -10+5=-5.

The period of the sine function is given by 2π/2/5=5π. This means that the function completes one full cycle every 5π units. To find the minimum and maximum values of the function, we need to evaluate it at the critical points of the function, which occur at intervals of 5π.

At x=0, the function has a value of 5+10sin(0)=15, which is the maximum value of the function. At x=5π/2, the function has a value of 5+10sin(2π/5)=5, which is the minimum value of the function.

At x=[tex]5π[/tex], the function has a value of 5+10sin(4π/5)=-5, which is the maximum value of the function. At x=15π/2, the function has a value of 5+10sin(6π/5) =5, which is the minimum value of the function.

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.m The correct equation to determine the amount of material used, m, to construct the fully enclosed tent is:

                         c. =2(8∙9)+2(7.5∙8)+(12∙9∙6)

To determine the amount of material used to construct the fully enclosed tent, we need to consider the surface area of the tent. The tent is fully enclosed, so we need to calculate the area of all the sides.

Option a. =(8∙9)+2(7.5∙8) calculates the area of the top and two sides of the tent. This does not include the front and back of the tent, so it is not the correct equation.

Option b. =(8∙9)+2(7.5∙8)+2(12∙9∙6) calculates the area of the top, two sides, front and back of the tent, but it also includes an extra term of 2(12∙9∙6) which is not necessary for a fully enclosed tent. This option overestimates the amount of material used.

Option c. =2(8∙9)+2(7.5∙8)+(12∙9∙6) calculates the area of the top, bottom, and all four sides of the tent. This is the correct equation to determine the amount of material used in a fully enclosed tent.

Option d. =3(8∙9)+2(12∙9∙6) overestimates the amount of material used because it includes an extra term of 3(8∙9) which is not necessary for a fully enclosed tent.

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