Question 574·Hard·Math

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A lidless box with a square base is to be constructed using $1{,}200$ square centimeters of material for the base and four side faces. If the side length of the square base is $x$ centimeters, which function $V$ gives the volume of the box, in cubic centimeters, in terms of $x$ ?

A surface-area equation describes how much material covers a box; a volume equation describes how much space it holds. Count only the faces being built, solve the material equation for the height, then multiply the base area by that height. The key traps are adding a lid that isn't there and forgetting that each wall has both a width and a height.

Hints

Hint 1
The material covers the square base and four walls, but no lid. A square with side $x$ has area $x^2$ . If the box has height $h$ , what is the area of one wall?
Hint 2
To find the height, first subtract the base's area from the total material. The area left covers four walls, each $x$ centimeters wide. What must you divide by?
Hint 3
Volume is base area times height. Once you have $h$ in terms of $x$ , put it into $V=x^2h$ and look for a factor you can cancel.

Step-by-step

Find the height from the material
Step 1
Account for every face
No Desmos needed. The answer is a formula in $x$ , so keep the lengths symbolic. Let $h$ be the box's height. The surface area of the square base is $x^2$ . Each of the four side faces has area $xh$ . There is no lid, so the material equation is:
$x^2+4xh=1{,}200.$
Step 2
Remove the base area
Subtract $x^2$ from both sides to leave the material used for the walls:
$4xh=1{,}200-x^2.$
Step 3
Find the height
Because $x$ is a positive side length, divide by $4x$ :
$h=\frac{1{,}200-x^2}{4x}.$
Step 4
Write the volume rule
Volume is base area times height. The base area is $x^2$ , so:
$V(x)=x^2h.$
Step 5
Put the height into the volume rule
Substitute the height you found:
$V(x)=\frac{x^2(1{,}200-x^2)}{4x}.$
Step 6
Simplify the function
Cancel one factor of $x$ from the top and bottom: $V(x)=\frac{x(1{,}200-x^2)}{4}$ . This gives the box's volume in cubic centimeters for a base side length of $x$ centimeters. Choice D.

Question 574·Hard·Math

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