A volume in cubic units calls for multiplying a rectangular solid’s three dimensions. When those dimensions are described in terms of one another, name the requested width first and write the other lengths from it. Then solve the volume equation with a Desmos regression, keeping only a width that makes the height positive. The trap is reporting the length or height instead of the width.
Hints
- Hint 1
Let be the width. Twice the width means for the length; inches less than the width means for the height. Write all three dimensions before using the volume.
- Hint 2
A rectangular solid is a box. Its volume is length times width times height, not their sum. What product of your three expressions must equal the given volume?
- Hint 3
In a Desmos regression, use for the unknown and in place of the equals sign. Keep , since the height must be positive.
Step-by-step
Build the volume equation and solve in Desmos
Step 1Write each dimension using the width
Let be the width. Write every dimension in terms of the width: twice the width makes the length , and inches less than the width makes the height , not . A height must be positive, so .
- Step 2
Turn the volume into an equation
A rectangular solid’s volume is length times width times height. The given volume is cubic inches, so multiply all three dimensions: .
- Step 3
Solve for the allowed width
Type , replacing with and with . Desmos uses the regression to find a value that makes both sides equal; the restriction keeps the height positive. Under PARAMETERS, it shows . Since represents the width, the solid is inches wide. Grid in 6.