Any and all help would be greatly appreciated. I'm trying to help my niece and I'm afraid that it is not possible.

The volume of a cubic container is x^3 cubic inches. The height of the container was decreased and the length was increased so that the volume is now modeled by the expression,
x^3+4x^2-21x

By how many feet were the height and lenght changed.

To determine by how many units the height and length were changed, we need to find the difference between the original volume expression (x^3 cubic inches) and the new volume expression (x^3 + 4x^2 - 21x cubic inches).

Let's subtract the original volume expression from the new volume expression:

(x^3 + 4x^2 - 21x) - (x^3)

Simplifying this expression, we get:

x^3 + 4x^2 - 21x - x^3

The x^3 terms will cancel each other out:

4x^2 - 21x

Now, let's factor out the common term:

x(4x - 21)

This expression represents the change in volume. To determine the change in height and length, we need to find the value of x that makes this expression equal to zero.

Setting (4x - 21) equal to zero:

4x - 21 = 0

Adding 21 to both sides:

4x = 21

Dividing both sides by 4:

x = 21/4

Therefore, the value of x is 5.25.

This means that the height and length were changed by approximately 5.25 units.

To determine the change in height and length of the container, we need to find the difference between the original dimensions and the new dimensions modeled by the given expression.

Let's break down the expressions representing the original and new volumes:

Original volume: x^3 cubic inches
New volume: x^3 + 4x^2 - 21x cubic inches

The original volume can be represented as a cube with a side length of x inches.

The new volume can be represented as a rectangular prism with its dimensions expressed in terms of x.

To find the change in height and length, we need to compare the dimensions of the cube and the rectangular prism.

Let's equate the volumes and solve for the height and length of the rectangular prism:

x^3 = x^3 + 4x^2 - 21x

Simplifying the equation, we get:

0 = 4x^2 - 21x

Now, let's solve for x:

4x^2 - 21x = 0

Factor out x:

x(4x - 21) = 0

Setting each factor equal to zero:

x = 0 (we disregard this solution since it represents a volume of zero)

4x - 21 = 0

4x = 21

x = 21/4

Therefore, the value of x is 21/4.

Now that we have the value of x, we can substitute it back into the expressions for the original and new volumes to calculate their dimensions.

For the original volume (x^3), the side length of the cube is:

original side length = x = 21/4 inches

For the new volume (x^3 + 4x^2 - 21x), the dimensions of the rectangular prism are given by:

length = x = 21/4 inches
height = 4x^2 - 21x = (4(21/4)^2) - (21(21/4)) = 441/4 - 441/4 = 0 inches

Therefore, the change in height and length is 0 feet.