Question

An open box with no lid has a square base and four sides of equal height. The height is 4 inches

greater than the length and width (which are the same). What are the dimensions of the box if the
volume is 63 cubic inches and the surface area is 93 square inches?
PLEASE SHOW YOUR WORK:) THANK YOU SO MUCH

Answer 1

width = length = 3 inches

height = 7 inches

Explanation:

If x is the width and length of the base, and y is the height, then:

y = x + 4

The volume of the box is:

63 = x²y

The surface area of the box is:

93 = x² + 4xy

Substitute the first equation into the third.

93 = x² + 4x (x + 4)

93 = x² + 4x² + 16x

0 = 5x² + 16x − 93

0 = (x − 3) (5x + 31)

x = 3

y = 7

Use the second equation to check your answer.

63 = (3)²(7)

63 = 63

Answer 2

Length=Width=3

Height=7.

Explanation:

First, let's write some equations. So, we have an open box (with no lid) that has a square base. It has a height 4 units more of its width/length.

First, let's write the equation for the volume. The volume of a rectangular prism is:

`V=lwh`

Recall that we have a square base. In other words, the length and width are exactly the same. Therefore, we can do the following substitution:

`V=(w)wh=w^2(h)`

Now, recall that the height is four units more than the width/length. Therefore, we can make the following substitution:

`V=w^2(w+4)\\63=w^2(w+4)`

We can't really do anything with this. Let's next find the equation for the surface area.

So, we have 5 sides (not 6 because we have no lid). The bottom side is a square, so it's area is w^2. Since we have a square base, the remaining four sides will have an area w(w+4). In other words:

`93=w^2+4(w(w+4))`

The left term represents the area of the square base. The right term represents the area of one of the rectangular sides, times sides meaning four sides. Simplify:

`93=w^2+4w^2+16w\\5w^2+16w-93=0`

This seems solvable. Let's try it. Trying factoring by guessing and checking.

We can see that it is indeed factor-able. -15 and 31 are the numbers:

`5w^2-15w+31w-93=0\\5w(w-3)+31(w-3)=0\\(5w+3)(w-3)=0\\w=3\\h=w+4=7`

We ignore the other one because width cannot be negative.

So, the width/length is 3 and the height is 7. We can check this by plugging this into the volume formula:

`63\stackrel{?}{=}(3)^2(7)\\63\stackrel{\checkmark}{=}63`