The radius of a circle is multiplied by 1
4
. Which of the following describes the effect of this change on the area?
a. The area is multiplied by 1
16
.
b. The area is multiplied by 1
4
.
c. The area is multiplied by 1
8
.
d. The area is multiplied by 4
.

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.

The correct research hypothesis and distribution is directional, t-test for independent means.

The research hypothesis that "people can hear better when they have just eaten a large meal" is a directional hypothesis because it predicts the direction of the effect (i.e., hearing ability will improve after a large meal).

The appropriate statistical test to use would be a t-test for independent means, which compares the means of two independent groups to determine if there is a statistically significant difference between them.

Therefore, the correct answer is B. Directional; t-test for independent means.

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The probability of obtaining a reading greater than 0.28°C is 0.3897.

What is standard normal distribution ?

The standard normal distribution is a specific type of probability distribution that has a mean of 0 and a standard deviation of 1. It is also called the Z-distribution or the Gaussian distribution.

The standard normal distribution is commonly used in statistics and probability theory to make comparisons and calculations across different normal distributions. To use the standard normal distribution for calculations involving a normal distribution with a different mean and standard deviation, the data must be standardized by subtracting the mean and dividing by the standard deviation.

According to the question:
To solve this problem, we need to standardize the value of 0.28°C using the standard normal distribution formula:

z = (x - mu) / sigma

where:

x = 0.28°C

mu = 0°C

sigma = 1.00°C

Substituting the values, we get:

z = (0.28 - 0) / 1.00

z = 0.28

Now, we need to find the probability of obtaining a reading greater than 0.28°C, which is the same as finding the area to the right of z = 0.28 on the standard normal distribution curve. We can use a standard normal distribution table or calculator to find this area.

Using a calculator or software, we find that the probability of obtaining a reading greater than 0.28°C is 0.3897, rounded to 4 decimal places.

Therefore, the probability of obtaining a reading greater than 0.28°C is 0.3897.

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P = 2 is the answer to the equation 2(p + 4) = 12.

An expression is made up of a number, a variable, or a combination of a number, a variable, and operation symbols. Two expressions are combined into one equation by using the equal symbol. For illustration: When you add 8 and 3, you get 11.

Divide the two among the terms between the parenthesis:

2p + 8 = 12

Add 8 to both sides of the equation, then subtract 8:

2p + 8 - 8 = 12 - 8

2p = 4

multiply both sides by two:

2p/2 = 4/2 \sp = 2

p = 2 is the answer to the equation 2(p + 4) = 12 as a result.

Simply put p = 2 back into the equation and simplify to obtain p + 4:

[tex]p + 4 = 2 + 4 = 6[/tex]

Hence, p + 4 = 6.

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The homotopy classes of paths from x to x in a space X refer to a set of equivalence classes of continuous paths that start and end at the same point, x, in the space X, where equivalence is defined in terms of homotopy.

In other words, for any two paths, there exists a continuous transformation (called a homotopy) between them such that the endpoints remain fixed. Two paths are said to be homotopic if they can be continuously deformed into each other while keeping their endpoints fixed. The set of all paths that are homotopic to each other forms an equivalence class.

The homotopy classes of paths from x to x are important in algebraic topology, as they provide a way to study the topological structure of a space by analyzing the properties of the paths within it. They can also be used to define higher algebraic structures such as the fundamental group and higher homotopy groups.

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The shear force and bending moment diagrams for the given beam will have multiple segments of different shapes and slopes, reflecting the variation of loads along the length of the beam.

To construct the shear and bending diagrams for the given beam, we need to analyze the beam for the different sections where the load is applied. We can break down the beam into five sections:

Leftmost section (0 ≤ x ≤ L/4)

Second section (L/4 < x ≤ L/2)

Third section (L/2 < x ≤ 5L/12)

Fourth section (5L/12 < x ≤ 7L/12)

Rightmost section (7L/12 < x ≤ L)

We can use the equations for shear and bending moments to create the plots:

For section 1: 0 ≤ x ≤ L/4

The shear force diagram will be constant since there is no load applied in this section. The bending moment diagram will be a sloping line, which will be zero at x = 0 and will increase linearly with x as we move toward the right end of the section.

For section 2: L/4 < x ≤ L/2

The shear force diagram will start from the value of P at x = L/4 and remain constant up to x = L/2. The bending moment diagram will be a parabolic curve, which will be zero at x = L/4 and L/2 and will reach a maximum value at the midpoint of the section.

For section 3: L/2 < x ≤ 5L/12

The shear force diagram will start from the value of P at x = L/4 and remain constant up to x = L/2. At x = 5L/12, a load of P/3 is added, causing the shear force to increase suddenly. The bending moment diagram will be a cubic curve, which will be zero at x = L/4 and L/2, and will have a local minimum at x = 5L/12.

For section 4: 5L/12 < x ≤ 7L/12

The shear force diagram will start from the value of P + P/3 at x = 5L/12 and remain constant up to x = 7L/12. The bending moment diagram will be a cubic curve, which will be zero at x = L/4 and L/2, and will have a local maximum at x = 7L/12.

For section 5: 7L/12 < x ≤ L

The shear force diagram will start from the value of P + P/3 at x = 5L/12 and remain constant up to x = 7L/12. At x = L/3, a load of P/3 is added, causing the shear force to decrease suddenly. The bending moment diagram will be a parabolic curve, which will be zero at x = L/4 and L/2 and will reach a minimum value at the midpoint of the section.

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Answer:

3n squared+n+2

Answer:

the nth term of the sequence is 2n² + 4T

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Step-by-step explanation:

he sequence represents a quadratic function, the nth term of the sequenceis 2n² + 4 

The nth term of a quadratic sequence is :an² + bn + c c = zeroth term ; a = second difference ÷ 2

From the sequence :First difference = 6, 10, 14, 18Second difference = 4, 4, 4

 First difference between terms in position 1 and 0 :6 - 4 = 2 

Zeroth term = First term in sequence - 2 = 6 - 2 = 4a = 4/2 = 2

Plugging the values into the equation :2n² + bn + 4 Using the 2nd term :n = 22(2)² + 2b + 4 = 128 + 2b + 4 = 12 2b = 12 - 12 2b = 0b = 0

Hence, the nth term of the sequence is 2n² + 4

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The workload of the resource is 20.5 units.

To calculate the workload of a resource that serves different flow unit types, one must know the amount of flow units, the processing time for each flow unit, and the number of resources available. This is best calculated using Little's Law, which states that the average number of flow units in a system is equal to the average rate of flow units multiplied by the average time they spend in the system.

For example, if a resource is serving 3 flow unit types, A, B and C, with 10, 8 and 5 units respectively, and a processing time of 2 minutes, 1 minute and 3 minutes respectively, with 2 resources available, the workload can be calculated as follows:

Workload = (10*2 + 8*1 + 5*3) / 2

       = 41 / 2

       = 20.5 units

Therefore, the workload of the resource is 20.5 units.

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Complete question

What are the flow unit types that the resource is serving?

Answer:

A) 4

Step-by-step explanation:

There are 2 lines dividing the polygon.

The first two on the top are triangles.

The two on the bottom are irregular polygons.

Added up, that's 4 polygons according to the line distribution.

Answer:

125

Step-by-step explanation:

We Take

625 / 5 = 125

So, the answer is 125

The length of the base of the triangle is 8 yards whose height is half of 28 yards.

A triangle is a two-dimensional geometric shape that is formed by three straight lines that connect three non-collinear points. These three lines are called the sides of the triangle, and the points where the sides meet are called the vertices of the triangle.

The following formula provides the area of a triangle:

Area = (1/2) x base x height

We are given that the height of the triangle is half of 28 yards, which is:

height = 1/2 x 28 = 14 yards

We are also given that the area of the triangle is 56 square yards. Substituting these values into the formula for the area, we get:

56 = (1/2) x base x 14

Simplifying this equation, we get:

56 = 7 x base

Dividing both sides by 7, we get:

base = 8

Therefore, the length of the base of the triangle is 8 yards.

The vertices are typically denoted by letters, such as A, B, and C. The three angles formed by the sides are also part of the triangle.

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Answer:

I don't knowI am sorry I will let someone else answer

Step-by-step explanation:

The value of x in the given equation is 15/23.

Two expressions are combined in an equation by an equal symbol ("="). The "left-hand side" and "right-hand side" of the equation are the two expressions on either side of the equals symbol. Typically, we consider an equation's right side to be negative. Since we can balance this by deducting the right-side expression from both sides' expressions, this won't decrease the generality.

in the given equation, 12x+6y=12y=x+15

12x+6y=12y

2x+y=2y

y=2x

now we have,

12y=x+15

12(2x)=x+15

24x=x+15

x=15/23

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The middle 50% data of the boxplot will be calculated between the value of 7 to 14.

The data set ranges are:

4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

Divide the given data in 4 quartiles Qs;

Q1 = 4 - 6

Q2 = 7 - 10

Q3 = 11 - 14

Q4 = 15 - 17

Thus, 50% data will be lying in Q2 and Q3.

Range - 7 - 14

Thus, the middle 50% data of the box plot will be calculated between the value of 7 to 14.

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Answer:

the answer is $10

Step-by-step explanation:

trust that

Answer: The estimated product is about 10.

Step-by-step explanation:

To estimate the product, we can round 2 1/6 to 2 and 4 1/2 to 5. Then, we multiply 2 x 5 to get an estimated product of 10.

To find the exact product, we can use the following steps to multiply the two mixed numbers:

   Convert each mixed number to an improper fraction:

   2 1/6 = 13/6

   4 1/2 = 9/2

   Multiply the two fractions:

   (13/6) x (9/2) = (13 x 9) / (6 x 2) = 117/12

   Simplify the fraction, if possible:

   117/12 = 9 3/4

Therefore, the exact product of 2 1/6 x 4 1/2 is 9 3/4.

Accοrding tο the data, the answers tο Questiοns 1 and 2 are: Harris shοuld anticipate spinning a sum οr less 10 apprοximately 367 times and a tοtal that is 10 οr larger apprοximately 133 times.

Yοur example is an equatiοn since an equatiοn that's any statement with an equal's sign. The usage οf equatiοns in mathematical expressiοns is widespread because mathematicians adοre equal signs. An equatiοn is a cοllectiοn οf guidelines fοr prοducing a specific οutcοme.

Part A: Tο calculate the likelihοοd that Harris will spinning a sum οr less 10, multiply the οverall number οf spins by the chance οf οbtaining a sum οr less 10. The οutcοmes οf the first spinner's spin are 1, 2, and 3, while the results οf the secοnd spinner's spin are 4, 5, 6, 7, and 8. Hence, the amοunts οr less 10 are:

1 + 4 = 5

1 + 5 = 6

1 + 6 = 7

1 + 7 = 8

1 + 8 = 9

2 + 4 = 6

2 + 5 = 7

2 + 6 = 8

2 + 7 = 9

3 + 4 = 7

3 + 5 = 8

3 + 6 = 9

There are 11 amοunts that cοuld be less than ten. The number οf successful results divided by the entire number οf pοssibilities, which is 11/15, represents the likelihοοd οf receiving a payοut οf less than 10 in a single spin. Harris will therefοre spin a tοtal less than 10 times, and the equatiοn tο estimate this is:

11/15 = x/500

After finding x, we οbtain:

x = (11/15) x 500

x = 366.67, which rοunds up tο 367

Sο, Harris shοuld expect tο spin a sum less than 10 abοut 367 times.

Part B: Tο determine hοw frequently Harris shοuld anticipate spinning a sum οf ten οr mοre, we can deduct the times that he shοuld anticipate spinning a sum lοwer than ten frοm the οverall number οf spins:

500 - 367 = 133

Therefοre, Harris shοuld expect tο spin a sum that is 10 οr greater abοut 133 times.

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Answer:

To write an exponential decay function for this situation, we can use the formula:

P(t) = P₀e^(rt)

where:

P(t) = the population at time t

P₀ = the initial population

r = the annual rate of decrease (as a decimal)

t = time in years

We are given P₀ = 3,846 and r = -0.0027 (since the population is decreasing).

To approximate the population in 2022, we need to find t, the number of years from 1996 to 2022. That is:

t = 2022 - 1996 = 26 years

Now we can plug in the values we have:

P(t) = 3,846 e^(-0.0027t)

To find P(2022), we plug in t = 26:

P(26) = 3,846 e^(-0.0027(26))

≈ 3,200.62

Therefore, we can approximate the population of the city in 2022 to be about 3,201 people.

Answer:

3,101

Step-by-step explanation:

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To write an exponential decay function for the population of the city, we can use the formula:

P(t) = P₀e^(-rt)

where P(t) is the population at time t, P₀ is the initial population, r is the decay rate, and e is the base of the natural logarithm.

In this problem, P₀ = 3,846 and r = 0.0027 (0.27% expressed as a decimal). We want to find the population in 2022, which is 26 years after 1996.To use the formula, we need to convert 26 years to the same time units as the decay rate. Since the decay rate is per year, we can use 26 years directly. Therefore, the exponential decay function for the population is:

P(t) = 3,846e^(-0.0027t)

To find the population in 2022 (t = 26), we substitute t = 26 into the function:

P(26) = 3,846e^(-0.0027*26) ≈ 3,101

Therefore, the population in 2022 is approximately 3,101.

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Answer:

28.05$

Step-by-step explanation:

The game got a discount of 15%.
New price is   (30$) ( 0.85 )  =  25.5$   (since the discount is 15% you only pay for the 85% of the original price)

Then an increase of 10%. this is:
New price:   25.5$ (1.10) = 28.05

New price is 28$ with 5 cents

The answer will of given mathematical logic will be T F T T F respectivelly.

A negation, conjunction, and disjunction are the fundamental mathematical logics. The symbols for negation, conjunction, and disjunction in mathematical logic are "," "," and "v," respectively.

Logical proofs frequently employ mathematical logic. Proofs are legitimate arguments that establish the veracity of mathematical assertions. A series of statements make up an argument. The conclusion is the last assertion, and the premises are all the statements that came before it (or hypothesis).

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The complete truth table is shown in the below diagram.

≈ q → ≈ p: True  False  True  True

The contrapositive of a conditional statement is a new conditional statement that is formed by negating both the hypothesis (the "if" part) and the conclusion (the "then" part) of the original statement, and switching their positions. The truth table for the contrapositive of a conditional statement has the same number of rows as the truth table for the original statement.

For example, if the original statement is "If it is raining, then the ground is wet", then the contrapositive would be "If the ground is not wet, then it is not raining."

According to the given table the contrapositive of a conditional statement q and p is defines as;

True

False

True

True

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The most likely to produce invalid statistical analysis of numeric measures is Pain rating as: none = 0; slight = 1; much = 2,  as it is an ordinal scale of measurement, which does not have equal intervals between the categories. So, the correct answer is B).

It means that the differences between the categories are not necessarily equivalent, and therefore, any statistical analysis based on this scale may not accurately reflect the true relationship between variables.

The other options (blood pressure in mmHg, oxygen saturation in percentage, neonatal birth weight in kilograms) are measured on interval or ratio scales, which have equal intervals between values and can be used for meaningful statistical analysis. so, the correct option is B).

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____The given question is incomplete, the complete question is given below:

Which of the following numeric measures would be most likely to produce invalid statistical analysis?A)Analysis of patients' blood pressures in mmHgB)Pain rating as: none = 0; slight = 1; much = 2C)Assessment of oxygen saturation in percentageD)Analysis of neonatal birthweight in kilogr

The simplification of the given expression

[tex]y = ( \sqrt{47 + \sqrt{9 - \sqrt{25)} } } [/tex]

is y = 7

[tex]y = ( \sqrt{47 + \sqrt{9 - \sqrt{25)} } } [/tex]

find the square root of 25

[tex]y =( \sqrt{47 + \sqrt{9 - 5} } [/tex]

simplify root 9 - 5

[tex]y = ( \sqrt{47 + \sqrt{4} } [/tex]

find the square root of 4

[tex]y = ( \sqrt{47 + 2)} [/tex]

Add root 47 and 2

[tex]y = ( \sqrt{49} )[/tex]

Find the square root of 49

y = 7

Therefore, the solution to the given expression is y = 7

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Answer:

Step-by-step explanation:

To solve this problem, we can use the formula for the Pythagorean theorem, which states that for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, we are given the length of two sides of the triangle (the legs) and we need to find the length of the hypotenuse.

Let's label the sides of the triangle:

The shorter leg is the vertical side opposite the angle marked 55 degrees, so let's call it "a".

The longer leg is the horizontal side adjacent to the angle marked 55 degrees, so let's call it "b".

The hypotenuse is the side opposite the right angle, so let's call it "c".

Using trigonometry, we can determine the value of "a" and "b":

a = b * tan(55°) (since tangent = opposite/adjacent, we solve for opposite which is "a" in this case)

a = 100 * tan(55°) = 100 * 1.428 = 142.8

b = 100

Now, we can use the Pythagorean theorem to find the length of the hypotenuse:

c^2 = a^2 + b^2

c^2 = 142.8^2 + 100^2

c^2 = 20484.84 + 10000

c^2 = 30484.84

c = sqrt(30484.84)

c ≈ 174.6

Therefore, the length of the hypotenuse is approximately 174.6 units (the units are not given in the problem, but we can assume they are consistent with the units used for the given values of "a" and "b").

The problem does not specify the orientation or scale of the graph, but we can assume that it is a right triangle with the angle marked 55 degrees in the upper left corner.

The vertical side (the shorter leg) of the triangle should be labeled with a length of approximately 142.8 units (assuming the units used for the problem are consistent with the values given for "a" and "b"). The horizontal side (the longer leg) should be labeled with a length of 100 units.

The hypotenuse (the side opposite the right angle) should be drawn as a diagonal line connecting the endpoints of the vertical and horizontal sides. The hypotenuse should be labeled with a length of approximately 174.6 units.

The angle marked 55 degrees should be labeled as such, and the other two angles of the triangle (the right angle and the angle opposite the longer leg) should be labeled accordingly.

Answer:

The answer is C

Step-by-step explanation:

Assuming this is a fair coin, the theoretical probability of the coin going on one side, let's say heads, is 50%, or 0.5. So what's the chance the coin lands head 5 times? To do this we do 0.5^5 OR 0.5*0.5*0.5*0.5*0.5. Both of these answers equal 0.03125. So C is the Answer. Hope this helps :D

Option C is the correct answer. 0.17 is the probability that a randomly selected student smokes out of 500 students, given 85 smoke.

By dividing the total number of students observed (500) by the number of students who smoke (85) in this scenario, it is possible to estimate the probability of smoking among the 500 students.

This results in a ratio of 0.17, or 17%. The estimated likelihood that a randomly chosen student smokes is therefore 0.17, meaning that roughly 17 out of every 100 students in the population smoke. It is crucial to remember that this is only an estimate, and the true probability could change slightly depending on the size and sampling method.

The estimated probability does, however, have a tendency to converge to the true probability when the sample size is sufficient.

Hence, the probability that a randomly selected student smokes  is 0.17

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Answer:

g

Step-by-step explanation:

The given expression is arcsin (-√3/2), which represents the angle whose sine is equal to -√3/2. Recall that the range of the arcsin function is from -π/2 to π/2 radians, so we can narrow down the possible solutions to the second and third quadrants.

Since the sine function is negative in the third quadrant, we can start by considering the angle 4π/3, which is in the third quadrant and has a sine of -√3/2:sin(4π/3) = -√3/2

However, we need to check if there are any other angles in the second or third quadrants that satisfy the equation. Recall that sine is periodic with a period of 2π, so we can add or subtract any multiple of 2π to the angle and still obtain the same sine value.

In the second quadrant, we can use the reference angle π/3 to find the corresponding angle with a negative sine:

sin(π - π/3) = sin(2π/3) = √3/2

This angle does not satisfy the equation, so we can eliminate it as a possible solution.In the third quadrant, we can use the reference angle π/3 to find another possible solution:

sin(π + π/3) = sin(4π/3) = -√3/2

This confirms our initial solution of 4π/3, so the answer is (g) 4π/3.

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Answer:

We know that sin(π/3) = √3/2, so we can write:

arcsin(-√3/2) = -π/3 + 2nπ or π + π/3 + 2nπ

where n is an integer.

Therefore, the values of arcsin(-√3/2) are:

a. π/3 + 2nπ

c. 11π/6 + 2nπ

e. 2π/3 + 2nπ

f. 7π/6 + 2nπ

So, options a, c, e, and f are all correct.

The mean bonus paid by large companies to their executives is $5 million and the modal class is 3.5-6.5.

To find the mean, we need to find the midpoint of each class and multiply it by the frequency, then add up all of these values and divide by the total frequency:

Class boundaries Midpoint Frequency Midpoint x Frequency

0.5-3.5 2                          11                                         22

3.5-6.5 5                          12                                        60

6.5-9.5 8                           4                                        32

9.5-12.5 11                           2                                       22

12.5-15.5 14                           1                                        14

Total 150  

Mean = (Midpoint x Frequency) / Total Frequency

Mean = 150 / 30

Mean = 5

Therefore, the mean bonus paid by large companies to their executives is $5 million.

To find the modal class, we need to look for the class with the highest frequency. In this case, the class with the highest frequency is 3.5-6.5, with a frequency of 12. Therefore, the modal class is 3.5-6.5.

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The coordinate of fourth point of a parallelogram is (2,0)

We need to remember that the diagonals of a parallelogram intersect each other at a halfway point and the midpoint of each diagonal is the same.

A parallelogram is a geometric object with sides that are parallel to one another in two dimensions. It is a form of polygon with four sides (sometimes known as a quadrilateral) in which each parallel pair of sides have the same length. A parallelogram's adjacent angles add up to 180 degrees.

The midpoint formula

[tex]M=\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}[/tex]

AC=BD

We can find the coordinates of the fourth vertex  (x,y) through this procedure:

For x

[tex]\frac{5-2}{2}=\frac{0+x}{2}\\\\x=2\\\\for\ y \\\frac{2-2}{2}=\frac{y+2}{2}\\\\y=0[/tex]

Hence the coordinate of fourth point is (2,0)

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Answer:

1.066 (3 d.p.)

Step-by-step explanation:

The volume of the solid formed by revolving a region, R, around a vertical axis, bounded by x = a and x = b, is given by:

[tex]\displaystyle 2\pi \int^b_ar(x)h(x)\;\text{d}x[/tex]

where:

[tex]\hrulefill[/tex]

We want to find the volume of the solid formed by revolving a region, R, around the y-axis, where R is bounded by:

[tex]y=\dfrac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{3}}[/tex]

[tex]y=0[/tex]

[tex]x=0[/tex]

[tex]x=1[/tex]

As the axis of rotation is the y-axis, r(x) = x.

Therefore, in this case:

[tex]r(x)=x[/tex]

[tex]h(x)=\dfrac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{3}}[/tex]

[tex]a=0[/tex]

[tex]b=1[/tex]

Set up the integral:

[tex]\displaystyle 2\pi \int^{1}_0x \cdot\dfrac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{3}}\;\text{d}x[/tex]

Take out the constant:

[tex]\displaystyle 2\pi \cdot \dfrac{1}{\sqrt{2\pi}}\int^{1}_0x \cdot e^{-\frac{x^2}{3}}\;\text{d}x[/tex]

[tex]\displaystyle \sqrt{2\pi}\int^{1}_0x \cdot e^{-\frac{x^2}{3}}\;\text{d}x[/tex]

Integrate using the method of substitution.

[tex]\textsf{Let}\;u=-\dfrac{x^2}{3}\implies \dfrac{\text{d}u}{\text{d}x}=-\dfrac{2x}{3}\implies \text{d}x=-\dfrac{3}{2x}\;\text{d}u[/tex]

[tex]\textsf{When}\;x=0 \implies u=0[/tex]

[tex]\textsf{When}\;x=1 \implies u=-\dfrac{1}{3}[/tex]

Rewrite the original integral in terms of u and du:

[tex]\displaystyle \sqrt{2\pi}\int^{-\frac{1}{3}}_0x \cdot e^{u}\cdot -\dfrac{3}{2x}\;\text{d}u[/tex]

[tex]\displaystyle \sqrt{2\pi}\int^{-\frac{1}{3}}_0 -\dfrac{3}{2}e^{u}\; \text{d}u[/tex]

[tex]-\dfrac{3\sqrt{2\pi}}{2}\displaystyle \int^{-\frac{1}{3}}_0 e^{u}\; \text{d}u[/tex]

Evaluate:

[tex]\begin{aligned}-\dfrac{3\sqrt{2\pi}}{2}\displaystyle \int^{-\frac{1}{3}}_0 e^{u}\; \text{d}u&=-\dfrac{3\sqrt{2\pi}}{2}\left[ \vphantom{\dfrac12}e^u\right]^{-\frac{1}{3}}_0\\\\&=-\dfrac{3\sqrt{2\pi}}{2}\left[ \vphantom{\dfrac12}e^{-\frac{1}{3}}-e^0\right]\\\\&=-\dfrac{3\sqrt{2\pi}}{2}\left[ \vphantom{\dfrac12}e^{-\frac{1}{3}}-1\right]\\\\&=1.06582594...\\\\&=1.066\; \sf (3\;d.p.)\end{aligned}[/tex]

Therefore, the volume of the solid is approximately 1.066 (3 d.p.).

[tex]\hrulefill[/tex]

[tex]\boxed{\begin{minipage}{3 cm}\underline{Integrating $e^x$}\\\\$\displaystyle \int e^x\:\text{d}x=e^x(+\;\text{C})$\end{minipage}}[/tex]

The answer of the given question based on solving  13 to the power of 16 is 13 to the power of 16 is 3,947,868,257,259,789. and the simplified expression is 7 + 2ab + br.

A expression is a combination of symbols or values that represents a particular concept or computation.

In mathematics, an expression is a combination of numbers, variables, operators, and/or functions that can be evaluated to produce a numerical result.

an expression is a way to represent an idea, computation, or meaning using a set of symbols or words.

To solve 13 to the power of 16, we can start by multiplying 13 by itself 16 times:

13 × 13 = 169

169 × 13 = 2197

2197 × 13 = 28,561

28,561 × 13 = 371,293

371,293 × 13 = 4,826,389

4,826,389 × 13 = 62,748,857

62,748,857 × 13 = 815,730,721

815,730,721 × 13 = 10,604,807,473

10,604,807,473 × 13 = 137,858,491,849

137,858,491,849 × 13 = 1,792,160,390,737

1,792,160,390,737 × 13 = 23,303,986,079,681

23,303,986,079,681 × 13 = 303,305,489,096,753

303,305,489,096,753 × 13 = 3,947,868,257,259,789

Therefore, 13 to the power of 16 is 3,947,868,257,259,789.

As for the second expression, we can simplify it using the distributive property of multiplication:

2(2 + ab) + b(r + 3) = 4 + 2ab + br + 3b

Simplifying further, we can combine the constant terms:

2(2 + ab) + b(r + 3) = 7 + 2ab + br

So the simplified expression is 7 + 2ab + br.

To know more about Distributive property visit:

https://brainly.com/question/13818728

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