Question

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

Answer 1

Answer:

Length=Width=3

Height=7.

Step-by-step 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:

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

Answer 2

Answer:

width = length = 3 inches

height = 7 inches

Step-by-step 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