CPM Homework Help : APCALC Problem 4-12

Khi thinks that when the flags at right are rotated, the $\left(\text{volume of }A\right)+$ $\left(\text{volume of }C\right)=$ $\left(\text{volume of}\left(B+C\right)\right)$ .

Decide if Khi is correct. If he is incorrect, explain the error in his logic.

Hint:

Start by visualizing the terms in Khi's equation: What does Volume $A$ and Volume $C$ look like? What does Volume $\left(B + C\right)$ look like?

Answer:

Volume $A$ is cone with radius $4$ . Volume $C$ is a cylinder with radius $2$ . Each is a separate 3D solid whose volumes are added together after they are computed: $V = [\text{cone with } r = 4] + [\text{cylinder with } r = 2]$

Hint:

Try computing the volume of each:

$\text{Volume } A + \text{Volume } C = \text{cone} + \text{cylinder}$

$=\frac{1}{3}\pi r^{2}h \ + \ \pi r^{2}h$

$\text{Volume} \left(B + C\right) = \text{Big Cone} − \text{Small Cone}$

$=\frac{1}{3}\pi r^{2}h \ - \ \frac{1}{3}\pi r^{2}h$

Evaluate and compare.

Answer:

Volume $\left(B + C\right)$ is a single 3D solid that looks like a cone with its top cut off. To compute its volume, it must be dissected into pieces:
$V = [\text{large cone with } r = 6] − [\text{small cone with } r = 2]$

Use the eTool below to visualize this problem.
Click on the link to the right to view the full version of the eTool: Calc 4-12 eTool

Horizontal segment, with shaded right triangle labeled, A, above segment, with its short leg, labeled 3, on the right third of the segment, & vertical leg labeled, 4.

Horizontal segment, with horizontal rectangle labeled, C, above segment, with its bottom edge, labeled 3, on the right third of the segment, & right edge labeled 2, with shaded right triangle labeled, B, sharing its horizontal leg with top edge of rectangle, vertical leg aligned with right edge of rectangle, labeled 4.