A human head can be approximated as a sphere with a circumference of 60
centimeters. What is the approximate volume of a human head, rounded to the nearest 1,000
cubic centimeters?(GEOMETRY HELP)

The integral of the function expressed as sum of partial frictions is -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C.

First, factor out 2x from the denominator to obtain:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[(x + 3)/(2x)(x² - 4)] dx

Next, we use partial fractions to express the integrand as a sum of simpler fractions. To do this, we need to factor the denominator of the integrand:

2x(x² - 4) = 2x(x + 2)(x - 2)

Therefore, we can write:

(x + 3)/(2x)(x² - 4) = A/(2x) + B/(x + 2) + C/(x - 2)

Multiplying both sides by the denominator, we get:

x + 3 = A(x + 2)(x - 2) + B(2x)(x - 2) + C(2x)(x + 2)

Now, we need to find the values of A, B, and C. We can do this by equating coefficients of like terms:

x = A(x² - 4) + B(2x² - 4x) + C(2x² + 4x)

x = (A + 2B + 2C)x² + (-4A - 4B + 4C)x - 4A

Equating coefficients of x², x, and the constant term, respectively, we get:

A + 2B + 2C = 0

-4A - 4B + 4C = 1

-4A = 3

Solving for A, B, and C, we find:

A = -3/4

B = 7/16

C = -1/16

Therefore, the partial fraction decomposition is:

(x + 3)/(2x)(x² - 4) = -3/(4(2x)) + 7/(16(x + 2)) - 1/(16(x - 2))

The integral becomes:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

Integrating each term separately gives:

∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

= -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

where;

Therefore, the final answer is:

∫[(x + 3)/(2x³ - 8x)] dx = -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

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a. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. b. 20! = 20 x 19 x 18 x ... x 2 x 1. c. There are 16 zeros at the end of the decimal representation of (20!)2.

a. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

b. 20! = 20 x 19 x 18 x ... x 2 x 1. To write this in factored form, we can identify the prime factors of each number and write the product using exponents. For example, 20 = 2² x 5, so we can write 20! as:

20! = (2² x 5) x 19 x (2 x 3²) x 17 x (2² x 7) x 13 x (2 x 2 x 3) x 11 x (2³) x (3) x (2) x 7 x (2) x 5 x (2) x 3 x 2 x 1

Simplifying, we get:

20! = 2¹⁸ x 3⁸ x 5⁴ x 7² x 11 x 13 x 17 x 19

c. The number of zeros at the end of (20!)² in decimal form is determined by the number of factors of 10, which is equivalent to the number of factors of 2 x 5. Since there are more factors of 2 than 5 in the prime factorization of (20!)², we only need to count the number of factors of 5. There are four factors of 5 in the prime factorization of 20!, which contribute four factors of 10 to the square. Therefore, (20!)²ends in 8 zeros when written in decimal form.

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By using rational zeros theorem, we find that there are no real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, so we cannot factor f(x) over the real numbers.

To find the real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, we can use the rational zeros theorem, which states that any rational zeros of the function must have the form p/q, where p is a factor of the constant term (-35) and q is a factor of the leading coefficient (1).

The possible rational zeros of f are therefore ±1, ±5, ±7, ±35. We can then test each of these values using synthetic division or long division to see if they are zeros of the function. After testing all of the possible rational zeros, we find that none of them are actually zeros of the function.

Therefore, we can conclude that there are no real zeros of the function f(x) = x^3 - x^2 - 37x - 35.

However, we could factor it into linear and quadratic factors with complex coefficients using the complex zeros of f(x). But since the problem only asks for factoring over the real numbers, we can conclude that the factored form of f(x) is:

f(x) = x^3 - x^2 - 37x - 35 (cannot be factored over the real numbers)

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Answer:

Step-by-step explanation:

u got this

The scale factor of dilation of the shaded in gray to the shaded in black is 3

Given that

The preimage = shaded in gray

The image = shaded in black

From the graph, we have the following values on the image and the preimage

The preimage = shaded in gray = 4

The image = shaded in black = 12

The scale factor of dilation is then calculated as

Scale factor = shaded in black/shaded in gray

So, we have

Scale factor = 12/4

Evaluate

Scale factor = 3

Hence, the scale factor dilation is 3

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The statement that best describes the mode of selection depicted in the figure is (b) Directional Selection, changing the average color of population over time.

The Directional selection is a type of natural selection that occurs when individuals with a certain trait or phenotype are more likely to survive and reproduce than individuals with other traits or phenotypes.

In the directional selection of evolution, the mean shifts that means average shifts to one extreme and supports one trait and leads to eventually removal of the other trait.

In this case, one end of the extreme-phenotypes which means that the dark-brown rats are being selected for. So, over the time, the average color of the rat population will change.

Therefore, the correct option is (b).

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The given question is incomplete, the complete question is

The figure below shows the change of a population over time. which statement best describes the mode of selection depicted in the figure?

(a) Disruptive Official, favoring the average individual

(b) Directional Selection, changing the average color of population over time

(c) Directional selection, favoring the average individual

(d) Stabilizing Selection, changing the average color of population over time

Answer:

$31,142

Step-by-step explanation:

To convert a percentage into a decimal, you move the decimal two places to the left. 1354% converted into a decimal is 13.54.

$23,000 * 13.54 = $31,142

From the figure 1. Two line segments are LA and EP. 2. Two rays are EC and AH. 3. Two lines are b and AP.

A ray is a segment of a line with a single endpoint and unlimited length in a single direction. A ray cannot be measured in terms of length.

The ends of a line segment are two. These endpoints are included, along with every point on the line that connects them. A segment's length can be measured, while a line's length cannot.

A line is a collection of points that extends in two opposing directions and is endlessly long and thin.

From the given figure we observe that,

1. Two line segments are LA and EP.

2. Two rays are EC and AH.

3. Two lines are b and AP.

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The value of the x is 5√3 after we successfully do the application of the 30°-60°-90° Triangle theorem.

The 30°-60°-90° Triangle Theorem states that in such a triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is the product of the length of the hypotenuse and the square root of 3 divided by 2.

Using this theorem, we can write:

y = hypotenuse

Opposite of 30° angle = 5 = hypotenuse/2

Opposite of 60° angle = x = hypotenuse × (√(3)/2)

Solving for the hypotenuse in terms of y from the first equation, we get:

hypotenuse = 5×2 = 10

Substituting this value into the third equation, we get:

x = 10 × (√(3)/2) = 5 × √(3)

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Answer: 1.85

Step-by-step explanation:

So, the probability of Freddie getting 4 base hits in a row is approximately 0.0088, or 0.88%.

Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

by the question.

Assuming that each at-bat is independent of the others, the probability of Freddie getting a base hit in one at-bat is 0.305.

To find the probability that he gets 4 base hits in a row, we can use the multiplication rule for independent events. This rule states that the probability of two or more independent events occurring together is the product of their individual probabilities.

Therefore, the probability of Freddie getting 4 base hits in a row is:

0.305 x 0.305 x 0.305 x 0.305 = 0.0088 (rounded to four decimal places)

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a) An automorphism of a group G is a bijective map φ:G→G that preserves the group structure. That is, φ(ab) = φ(a)φ(b) and φ(a⁻¹) = φ(a)⁻¹ for all a, b ∈ G.

b) The set-theoretic composition φψ of any two automorphisms φ, ψ is an automorphism, as it preserves the group structure and is bijective.

c) The set Aut(G) of all automorphisms of G, with the operation of composition of maps, is a group. This is because it satisfies the four group axioms: closure, associativity, identity, and inverses. Therefore, Aut(G) is a group under composition of maps.

An automorphism of a group G is a bijective map φ:G→G that preserves the group structure, meaning that for any elements a,b∈G, we have φ(ab) = φ(a)φ(b) and φ(a⁻¹) = φ(a)⁻¹. In other words, an automorphism is an isomorphism from G to itself.

To show that the set-theoretic composition φψ is an automorphism, we need to show that it satisfies the two conditions for being an automorphism. First, we have

(φψ)(ab) = φ(ψ(ab)) = φ(ψ(a)ψ(b)) = φ(ψ(a))φ(ψ(b)) = (φψ)(a)(φψ)(b)

using the fact that ψ and φ are automorphisms. Similarly,

(φψ)(a⁻¹) = φ(ψ(a⁻¹)) = φ(ψ(a))⁻¹ = (φψ)(a)⁻¹

using the fact that ψ and φ are automorphisms. Therefore, φψ is an automorphism.

To show that Aut(G) is a group, we need to show that it satisfies the four group axioms

Closure: If φ,ψ∈Aut(G), then φψ is also in Aut(G), as shown above.

Associativity: Composition of maps is associative, so (φψ)χ = φ(ψχ) for any automorphisms φ,ψ,χ of G.

Identity: The identity map id:G→G is an automorphism, since it clearly preserves the group structure and is bijective. It serves as the identity element in Aut(G), since φid = idφ = φ for any φ∈Aut(G).

Inverses: For any automorphism φ∈Aut(G), its inverse φ⁻¹ is also an automorphism, since it is bijective and preserves the group structure. Therefore, Aut(G) is closed under inverses.

Since Aut(G) satisfies all four group axioms, it is a group under composition of maps.

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In response to the stated question, we may state that The margin of error function is equal to the highest mistake on the estimate.

In mathematics, a function is a connection between two sets of numbers in which each member of the first set (known as the domain) corresponds to a single element in the second set (called the range). In other words, a function takes inputs from one set and produces outputs from another. Inputs are commonly represented by the variable x, whereas outputs are represented by the variable y. A function can be described using an equation or a graph. The equation y = 2x + 1 represents a linear function in which each value of x yields a distinct value of y.v

(a) The population distribution must be assumed to be normal before generating a confidence interval.

(b) The margin of error with 90% confidence is provided by:

Error Margin = Z (/2) * (/n)

Where Z (/2) is the confidence level/2 crucial value, is the population standard deviation (unknown), and n is the sample size.

Error Margin = t (/2, n-1) * (s/n)

Where t (/2, n-1) is the critical value for the degrees of freedom /2 and n-1, and s is the sample standard deviation.

(d) The margin of error is equal to the highest mistake on the estimate.

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Answer:

Show your solution ( 3. ) C + 18 = 29

Step-by-step explanation:

To solve the equation C + 18 = 29, we want to isolate the variable C on one side of the equation.

We can start by subtracting 18 from both sides of the equation:

C + 18 - 18 = 29 - 18

Simplifying the left side of the equation:

C = 29 - 18

C = 11

Therefore, the solution to the equation C + 18 = 29 is C = 11.

With the help of the given Venn diagram, the answer of n(A∪B) is 44 respectively.

A Venn diagram is a visual representation that makes use of circles to highlight the connections between different objects or limited groups of objects.

Circles that overlap share certain characteristics, whereas circles that do not overlap do not.

Venn diagrams are useful for showing how two concepts are related and different visually.

When two or more objects have overlapping attributes, a Venn diagram offers a simple way to illustrate the relationships between them.

Venn diagrams are frequently used in reports and presentations because they make it simpler to visualize data.

So, we need to find:

A ∪ B

Now, calculate as follows:

The collection of all objects found in either the Blue or Green circles, or both, is known as A B. Its components number is:

8 + 7 + 14 + 6 + 1 + 8 = 44

n(A∪B) = 44

Therefore, with the help of the given Venn diagram, the answer of n(A∪B) is 44 respectively.

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Answer:

False.

Step-by-step explanation:

A concave polygon can never be a regular polygon as it can never be equiangular. Each side of a regular polygon must be the same length, and all interior angles must also be equal.

Step-by-step explanation:

40 cans/student   X   20 students  X  15 gram/can = 12 000 gm  = 12 kg

Answer:

father age 45 son age 0 this is answer

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.

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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to get the equation of any straight line, we simply need two points off of it, let's use those two in the picture below

[tex](\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{-4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-4}-\stackrel{y1}{2}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{(-3)}}} \implies \cfrac{-6}{3 +3} \implies \cfrac{ -6 }{ 6 } \implies - 1[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{- 1}(x-\stackrel{x_1}{(-3)}) \implies y -2 = - 1 ( x +3) \\\\\\ y-2=-x-3\implies {\Large \begin{array}{llll} y=-x-1 \end{array}}[/tex]

The critical values, the intervals of increasing or decreasing and the maximum and minimum points of the f(x) is (-1.5, -16), x < -1.5 and x = -1.5 and for b (4,6) and (2,10), (2,4).

A) Critical values

We will find out the critical value by solving for f ' (x) = 0

therefore, taking the derivative of given function we get,

f ' (x) = 4(2x) + 12 = 0

        = 8x + 12 = 0

therefore, 8x = -12

                 x = -12/8

                 x= -1.5

x = -1.5 is the only critical value in x-coordinate. Now to determine the y-coordinate, simply put the value of x in the function f(x) = 4x2 + 12x - 7

we get, f(-1.5) = 4(-1.5)2 + 12 (-1.5) - 7

                      = 4(2.25) - 18 - 7

                      = 9 - 25 = -16  

therefore, the critical value of the function f(x) = 4x2 + 12x - 7 is (-1.5, -16)

f(x) =x3 - 9x2 + 24x - 10.

Intervals of increasing and decreasing function is i.e. f decreases for

x < -1.5.

Therefore, f has minimum value at x = -1.5.

B) Critical values

We will find out the critical value by solving for f ' (x) = 0

therefore, taking the derivative of given function we get,

f '(x) = 3x2 - 9(2x) + 24

       = 3x2 - 18x + 24 = 0

therefore, 3 ( x2 - 6x + 8) = 0

   i.e x2 - 6x + 8 = 0

        (x-4) (x-2) = 0

So, x = 4 or x = 2 are the two critical values in x-coordinate. Now to determine the y-coordinate, simply put the values of x in the function f(x) =x3 - 9x2 + 24x - 10

we get, Substituting x = 4

f(4) = 43 - 9 (4)2 +24 (4) -10

     = 64 - 144 + 96 - 10

     = 6

Now, Substituting x = 2

f(2) = 23 - 9(2)2 + 24(2) - 10

     = 8 - 36 + 48 - 10

     = 10

Therefore, the critical values of the function f(x) =x3 - 9x2 + 24x - 10 are (4,6) and (2,10).

Intervals of increasing and decreasing functions is f decreases in (2,4).

therefore, f has minimum at x = 4 and maximum at x = 2.

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Complete question:

For each function determine: i) the critical values ii) the intervals of increasing or decreasing iii) the maximum and minimum points.

a. f(x) = 4x²+12x–7 (3 marks)

b. F(x) = x°-9x²+24x-10 (3 marks)

According to the Venn diagram the value of [tex]n(A ^ C \cap B ^ C) = {3}[/tex] so the number of elements in that set is 1.

What is Venn diagram ?

A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. It is usually represented as a rectangle or a circle for each set and the overlapping areas between them, showing the common elements that belong to more than one set. Venn diagrams are widely used in mathematics, logic, statistics, and computer science to visualize the relationships between different sets and help solve problems related to set theory.

According to the question:
To solve this problem, we first need to understand the notation used.

n(A) denotes the set A and the numbers within the braces {} indicate the elements in set A. For example, n(A)={7,4,3,9} means that the set A contains 7, 4, 3, and 9.

n(AnB) denotes the intersection of sets A and B, i.e., the elements that are common to both A and B. For example, n(AnB)={4,3} means that the sets A and B have 4 and 3 in common.

^ denotes intersection of sets

cap denotes the intersection of sets

Now, we need to find the elements that are common to sets A and C, and sets B and C. We can do this by taking the intersection of A and C, and the intersection of B and C, and then taking the intersection of the two resulting sets.

[tex]n(A ^ C) = n(A \cap C) = {3,9}[/tex]

[tex]n(B ^ C) = n(B\cap C) = {3,5}[/tex]

Now, we take the intersection of [tex]n(A ^ C)[/tex] and [tex]n(B ^ C)[/tex]:

[tex]n(A ^ C \cap B ^ C) = {3}[/tex]

Therefore, the answer is 1.

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The sample mean is 2.1, the sample variance is 212.5% and the standard deviation is 14.57%

a. The sample mean can be computed as the average of the quarterly percent total returns:

[tex](11.2 - 20.5 + 13.2 + 12.6 + 9.5 - 5.8 - 17.7 + 14.3) / 8 = 2.1[/tex]

So the sample mean is 2.1%, which can be interpreted as the average quarterly percent total return for the stock over the sample period.

b. The sample variance can be computed using the formula:

[tex]s^2 = sum((x - mean)^2) / (n - 1)[/tex]

where x is each quarterly percent total return, mean is the sample mean, and n is the sample size. Plugging in the values, we get:

[tex]s^2 = (11.2 - 2.1)^2 + (-20.5 - 2.1)^2 + (13.2 - 2.1)^2 + (12.6 - 2.1)^2 + (9.5 - 2.1)^2 + (-5.8 - 2.1)^2 + (-17.7 - 2.1)^2 + (14.3 - 2.1)^2 / (8 - 1) = 212.15[/tex]

So the sample variance is 212.15%. The sample standard deviation can be computed as the square root of the sample variance:

[tex]s = \sqrt(s^2) = \sqrt(212.15) = 14.57[/tex]

So the sample standard deviation is 14.57%.

c. To construct a 95% confidence interval for the population variance, we can use the chi-square distribution with degrees of freedom n - 1 = 7. The upper and lower bounds of the confidence interval can be found using the chi-square distribution table or calculator, as follows:

upper bound = (n - 1) * s^2 / chi-square(0.025, n - 1) = 306.05

lower bound = (n - 1) * s^2 / chi-square(0.975, n - 1) = 91.91

So the 95% confidence interval for the population variance is (91.91, 306.05).

d. To construct a 95% confidence interval for the standard deviation (as percent), we can use the formula:

lower bound = s * √((n - 1) / chi-square(0.975, n - 1))

upper bound = s * √((n - 1) / chi-square(0.025, n - 1))

Plugging in the values, we get:

lower bound = 6.4685%

upper bound = 20.1422%

So the 95% confidence interval for the standard deviation (as percent) is (6.4685%, 20.1422%).

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Answer:

See below.

Step-by-step explanation:

We can solve this linear programming problem using the simplex method. We will start by converting the problem into standard form

Maximize z = 3x₁ + 5x₂ + 0s₁ + 0s₂

subject to

x₁ - 5x₂ + s₁ = 35

3x₁ - 4x₂ + s₂ = 21

x₁, x₂, s₁, s₂ ≥ 0

Next, we create the initial tableau

Basis x₁ x₂ s₁ s₂ RHS

s₁ 1 -5 1 0 35

s₂ 3 -4 0 1 21

z -3 -5 0 0 0

We can see that the initial basic variables are s₁ and s₂. We will use the simplex method to find the optimal solution.

Step 1: Choose the most negative coefficient in the bottom row as the pivot element. In this case, it is -5 in the x₂ column.

Basis x₁ x₂ s₁ s₂ RHS

s₁ 1 -5 1 0 35

s₂ 3 -4 0 1 21

z -3 -5 0 0 0

Step 2: Find the row in which the pivot element creates a positive quotient when each element in that row is divided by the pivot element. In this case, we need to find the minimum positive quotient of (35/5) and (21/4). The minimum is (21/4), so we use the second row as the pivot row.

Basis x₁ x₂ s₁ s₂ RHS

s₁ 4/5 0 1/5 1 28/5

x₂ -3/4 1 0 -1/4 -21/4

z 39/4 0 15/4 3/4 105

Step 3: Use row operations to create zeros in the x₂ column.

Basis x₁ x₂ s₁ s₂ RHS

s₁ 1 0 1/4 7/20 49/10

x₂ 0 1 3/16 -1/16 -21/16

z 0 0 39/4 21/4 525/4

The optimal solution is x₁ = 49/10, x₂ = 21/16, and z = 525/4.

Therefore, the maximum value of z is 525/4, which occurs when x₁ = 49/10 and x₂ = 21/16.

The standard deviation that the student would have in order to be publicly recognized is given as 1.17

We would have to assume that the students score follows a normal distribution

This is given as

X ~ (μ, σ)

(μ, σ) are the mean and the standard deviation

1 - 12 percent =

0.88 = 88 percent

using the excel function given as NORMS.INV() we would find the standard deviations

=NORM.S.INV(0.88)

= 1.17498

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A student would have to score approximately 0.89 standard deviations above the mean to be in the top 12% and be publicly recognized.

we can use the empirical rule to estimate the number of standard deviations a student has to score above the mean to be in the top 12 percent, assuming it is a normal distribution

The empirical rule states that for a normal distribution:

we will use the complement rule since our aim is to find the number of standard deviations a student has to score above the mean to be in the top 12%.

The complement of being in the top 12% is being in the bottom 88%.

From the empirical rule, we have that 68% of the data falls within one standard deviation of the mean.

Therefore, the remaining 32% (100% - 68%) falls outside one standard deviation of the mean.

Since we want to find the number of standard deviations a student has to score above the mean to be in the bottom 88%, we can assume that the remaining 32% is split evenly between the two tails of the distribution.      

Applying the z-score formula:

z = (x - μ) / σ

The z-score for a cumulative area of 0.44 is approximately -0.89  found by looking up the z-score corresponding to the cumulative area of 0.44 (half of 0.88) in a standard normal distribution table.

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Answer: x = 5

Step-by-step explanation:

To solve for x, we can first add 13 to both sides to isolate the variable term:

5x - 13 + 13 = 12 + 13

Simplifying the left side and evaluating the right side:

5x = 25

Then, divide both sides by 5 to isolate x:

5x/5 = 25/5

Simplifying:

x = 5

Therefore, the solution to the equation 5x - 13 = 12 is x = 5.

To solve for x in the equation 5x-13=12, we want to isolate the variable x on one side of the equation. We can do this by adding 13 to both sides of the equation:

5x-13+13 = 12+13

Simplifying, we get:

5x = 25

Finally, we can solve for x by dividing both sides of the equation by 5:

5x/5 = 25/5

Simplifying, we get:

x = 5

Therefore, the solution to the equation 5x-13=12 is x = 5.

Answer:

[tex]14 {y}^{2} \sqrt{ {x}^{3} {z}^{9} }[/tex]

Step-by-step explanation:

[tex] \sqrt{196 {x}^{3} {y}^{4}{z}^{9} }= \sqrt{196} \times \sqrt{ {x}^{3} } \times \sqrt{ {y}^{4} } \times \sqrt{ {z}^{9} } \\ \sqrt{196} = 14 \\ \sqrt{ {x}^{3} } = {x}^{ \frac{3}{2} } \\ \sqrt{ {y}^{4} } = {y}^{2} \\ \sqrt{ {z}^{9}} = {z}^{ \frac{9}{2} }[/tex]

A fractional exponent is not necessarily simpler so just take out the 1st and 3rd parts of the term which simplify nicely:

[tex] \sqrt{196 {x}^{3} {y}^{4}{z}^{9} } = 14 {y}^{2} \sqrt{ {x}^{3} {z}^{9} } [/tex]

Answer:

A large equilateral triangle pyramid stands in front of the city's cultural center. Each side of the base measures 40 feet and the slant height of each lateral side of the pyramid is 50 feet.

A painter can paint 100 square feet of the pyramid in 18 minutes.

How long does it take the painter to paint 75% of the pyramid?

Step-by-step explanation:

The total surface area of the pyramid can be calculated using the formula for the lateral surface area of a pyramid:

Lateral surface area = (1/2) × perimeter of base × slant height

Since the base is an equilateral triangle, the perimeter is 3 times the length of one side:

Perimeter of base = 3 × 40 feet = 120 feet

Lateral surface area = (1/2) × 120 feet × 50 feet = 3000 square feet

To paint 75% of the pyramid, the painter needs to paint:

0.75 × 3000 square feet = 2250 square feet

Since the painter can paint 100 square feet in 18 minutes, the time required to paint 2250 square feet can be calculated as:

2250 square feet ÷ 100 square feet per 18 minutes = 225 ÷ 10 × 18 minutes = 405 minutes

Therefore, the painter would need 405 minutes or 6 hours and 45 minutes to paint 75% of the pyramid.

Step-by-step explanation:

Refer to pic..........