Question

a cone has a base radius of 3 centimeters and a height of 5 centimeters. a student correctly calculates its volume to be 15π. the student thinks that a simpler formula for the volume of a cone is v = πrh because π(3)(5)=15π. which statement explains the conditions for which the students claim would be true? a. the claim is true only when the height is 5. b. the claim is true only when the radius of the cone is 3. c. the claim is true regardless of the dimensions of the cone. d. the claim is true whenever the product of the base and height is 15.

Explanation:

Step1: Recall volume formula of cone

The correct volume formula of a cone is $V=\frac{1}{3}\pi r^{2}h$. The student's proposed formula is $V = \pi rh$.

Step2: Set the two - formulas equal

If $\frac{1}{3}\pi r^{2}h=\pi rh$, we can solve for the condition. First, divide both sides of the equation by $\pi h$ (assuming $h
eq0$). We get $\frac{1}{3}r = 1$, then $r = 3$. If we divide both sides by $\pi r$ (assuming $r
eq0$), we get $\frac{1}{3}h=1$, then $h = 3$. Another way is to rewrite the correct formula as $V=\frac{1}{3}\pi r\cdot rh$. If $V=\pi rh$, then $\frac{1}{3}r = 1$ (when comparing the two expressions). In general, if we set $\frac{1}{3}\pi r^{2}h=\pi rh$, cross - multiply to get $r^{2}h = 3rh$. Rearranging gives $r^{2}h-3rh=0$, factoring out $rh$ gives $rh(r - 3)=0$. Since $r
eq0$ and $h
eq0$ (non - degenerate cone), $r = 3$.

Step3: Analyze the condition

The student's formula $V=\pi rh$ will give the correct volume of the cone only when the radius $r$ of the cone is 3. When $r = 3$, the correct volume formula $V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\times3^{2}h=\pi rh$.

Answer:

B. The claim is true only when the radius of the cone is 3.