Question

A rectangular piece of cardboard is 15 inches longer than it is wide. A 5-inch square is cut from each corner, and the remaining piece is folded to form a box. If the volume of the box is 1,250 cubic inches, what is the length of the box?

A.25 inches
B.30 inches
C.35 inches
D.40 inches

Answer 1

If the volume of the box is 1,250 cubic inches, the length of the box is: C. 35 inches.

In Mathematics and Geometry, the volume of a rectangular prism can be calculated by using the following formula:

Volume of a rectangular prism = l × w × h

Where:

• l represents the length of a rectangular prism.
• w represents the width of a rectangular prism.
• h represents the height of a rectangular prism.

Since the height of this ectangular box has 5-inch squares cut out, its dimensions include:

l: (w + 5) inches.

w: (w - 10) inches.

h: 5 inches.

By substituting the given dimensions (parameters) into the formula for the volume of a rectangular prism, we have the following;

1250 = (w + 5)(w - 10)5

250 = (w + 5)(w - 10)

`250=w^2-5w-50\\\\w^2-5w-300=0\\\\(w + 15)(w - 20)=0`

w = 20 or w = -15

Since the width cannot be negative, it must be 20 inches.

Since the rectangular piece of cardboard is 15 inches longer than it is wide, we have:

l = w + 15

l = 20 + 15

l = 35 inches.

Answer 2

Let x be the width of the cardboard, then the length is x+15. If 5 unches square is cut from each corner, then the new length is x+15-5·2=x+5 and the new width is x-5·2=x-10. The height of the formed box is 5 in. Now you can find the box volume:

`V=\text{ width }\cdot \text{ length }\cdot \text{ height },\\ V=(x-10)\cdot (x+5)\cdot 5,\\ V=5(x^2-5x-50)`.

If the volume of the box is 1,250 cubic inches, then

`5(x^2-5x-50)=1250,\\ x^2-5x-50=250,\\ x^2-5x-300=0,\\ D=(-5)^2-4\cdot (-300)=25+1200=1225,\\ √(D) =35,\\ x_(1,2)=(5\pm 35)/(2) =20,-15`.

The length of the box is x+5=20+5=25 in. (the negative x is extra, because the length couldn't be negative).

Answer: the length of the box is 25 inches.