Cubic inches cue volume: for a box, multiply the area of its base by its height. If a base side is described as a multiple of the height, name the height first so every dimension uses one unknown. Enter the resulting one-variable equation as a Desmos regression. An open top changes surface area, not how much space the box holds.
Hints
- Hint 1
Let stand for the height. Since each base edge is three times the height, write the length of an edge in terms of rather than treating as an edge length. What is it?
- Hint 2
A square base has area side times side. Use your expression for the base edge twice, then multiply that area by the height to match the given volume. Does the missing lid change how much the box holds?
- Hint 3
Once every dimension is in terms of , you have one equation with one unknown. Change the equals sign to in Desmos so a regression can find the height.
Step-by-step
Write one volume equation
Step 1Express the base side using the height
Let be the height, measured from the base to the open top. Each side of the square base is 3 times that height, so each side is inches. The is a multiplier, not a 3-inch side.
- Step 2
Build the volume equation
A box’s volume is base area times height, and a square’s area is side times side. So use the base side twice and set the volume equal to the given cubic inches: . Square the whole base side before multiplying by the height. The missing lid doesn’t change the volume.
- Step 3
Find the height in Desmos
Type into Desmos. The starts a regression, which finds the value of that makes the two sides equal. Under PARAMETERS, Desmos shows . Since is the height, the box is inches tall. Choice C.