the variable j varies jointly as the square of h and as the cube root of a, and it varies inversely as s. What happens to j when h is doubled, a is multiplied by 8, and s is multiplied by 4?

j = k(h^2)(a^(1/3)/s

when h is doubled , a multiplied by 8, s multiplied by 4

j2 = k (4h^2)(2a^(1/3)/(4s)

j2 / j = [ k (4h^2)(2a^(1/3)/(4s) ] / [k(h^2)(a^(1/3)/s]
= 4(2)/2 = 2

the value of j will double

Let's break down the given information step by step:

1. The variable j varies jointly as the square of h and as the cube root of a.
This can be written as: j = kh^2 * ka^(1/3), where k is a constant.

2. The variable j varies inversely as s.
This can be written as: j = ks^(-1), where k is a constant.

Now, let's analyze what happens to j when h is doubled, a is multiplied by 8, and s is multiplied by 4:

When h is doubled:
- h is multiplied by 2.
- According to the first statement, j varies jointly as the square of h.
- Since h is squared, the new value of h^2 will be 4 times the original value.

When a is multiplied by 8:
- a is multiplied by 8.
- According to the first statement, j varies jointly as the cube root of a.
- Since a is raised to the power of 1/3, the new value of a^(1/3) will be 2 times the original value.

When s is multiplied by 4:
- s is multiplied by 4.
- According to the second statement, j varies inversely as s.
- Since s is raised to the power of -1, the new value of s^(-1) will be 1/4 times the original value.

Now, let's put everything together and find the new value of j:

j = kh^2 * ka^(1/3) * ks^(-1)

New j = k * (4h^2) * (2a^(1/3)) * (1/4s)

Canceling out the common factors:

New j = k * 2h^2 * a^(1/3) * 1/s

Simplifying further:

New j = 2k * h^2 * a^(1/3) / s

Therefore, when h is doubled, a is multiplied by 8, and s is multiplied by 4, the value of j is multiplied by 2k * h^2 * a^(1/3) / s.

To determine what happens to the variable j when h is doubled, a is multiplied by 8, and s is multiplied by 4, we need to analyze the given information about the joint and inverse variation.

Let's break down the problem statement and identify the relationships between the variables:

1. "The variable j varies jointly as the square of h and as the cube root of a": This means that j is directly proportional to the square of h and the cube root of a. Mathematically, we can write this as:

j ∝ h^2 * (a)^(1/3)

2. "It varies inversely as s": This means that j is inversely proportional to s. Mathematically, we can write this as:

j ∝ 1/s

Now, we can combine these relationships to find the equation that represents the variation of j in terms of h, a, and s:

j = k * (h^2 * (a)^(1/3)) / s

Where k is the constant of variation.

To analyze what happens to j when h is doubled, a is multiplied by 8, and s is multiplied by 4, we can substitute the new values into the equation and simplify.

Let's assume the initial values are h₀, a₀, and s₀, and the final values are h₁, a₁, and s₁:

h₁ = 2h₀ (h is doubled)
a₁ = 8a₀ (a is multiplied by 8)
s₁ = 4s₀ (s is multiplied by 4)

Now, let's substitute these values into the equation:

j₁ = k * ((2h₀)^2 * (8a₀)^(1/3)) / (4s₀)

Simplifying this expression:

j₁ = k * (4h₀^2 * 2 * (2a₀)^(1/3)) / (4s₀)
j₁ = k * [(2^2 * h₀^2 * (2a₀)^(1/3)) / (4s₀)]
j₁ = k * (h₀^2 * (2a₀)^(1/3)) / s₀

We can see that j₁ is equal to j₀ (initial value of j), multiplied by a constant factor of (h₀^2 * (2a₀)^(1/3)) / s₀.

Therefore, when h is doubled, a is multiplied by 8, and s is multiplied by 4, the variable j will also be multiplied by the factor (h₀^2 * (2a₀)^(1/3)) / s₀.