Question

A cookie company packages its cookies in rectangular prism boxes designed with square bases that have both a length and width of 4 inches less than the height of the box. Find the dimensions of the box if its volume is 128 cubic inches.

Answer 1

The dimensions of the rectangular prism box with a square base, with each side being 4 inches less than the height and a total volume of 128 cubic inches, are found to be 4 inches × 4 inches × 8 inches.

Step-by-step explanation:

The student has been asked to find the dimensions of a box, where the base of the box is a square with sides 4 inches less than the height. Given that the volume of the box is 128 cubic inches, let's denote the height of the box as 'h' inches. Then the length and width of the base would be 'h-4' inches each. Therefore, the volume of the box can be expressed as the product of its dimensions:

Volume = length × width × height

Substituting the given terms, we have:

Volume = (h-4) × (h-4) × h

We are given that the volume is 128 cubic inches:

128 = (h-4) × (h-4) × h

To find the value of h, we need to solve the cubic equation. Upon solving, we find that h = 8 inches. Thus, the dimensions of the box are:

• Length = h - 4 = 8 inches - 4 inches = 4 inches
• Width = h - 4 = 8 inches - 4 inches = 4 inches
• Height = h = 8 inches

So, the dimensions of the rectangular prism box with a square base and a volume of 128 cubic inches are Length × Width × Height = 4 inches × 4 inches × 8 inches.

Answer 2

length and width=4

height=8

Step-by-step explanation:

Hello to solve this problem we must propose a system of equations of 3x3, that is to say 3 variables and 3 equations.

Ecuation 1

Leght=Width

.L=W

Ecuation 2

To raise the second equation we consider that the length and width of 4 inches less than the height of the box

H-4=W

Ecuation 3

To establish equation number 3, we find the volume of a prism that is the result of multiplying length, width, and height

LxWxH=128

From ecuation 1(w=h)

`HW^2=128`

solving for H

`H=(128)/(W^2)`

Using ecuation 2

H-4=W

`(128)/(W^2)-4=W\\W^3+4W^2-128=0`

Now we find the roots of the equation, 2 of them are imaginary, and only one results in 4

W=4in

L=4in

to find the height we use the ecuation 2

H-4=W

H=4+W

H=4+4=8

H=8IN