Grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height. How high is the pile (to the nearest hundredth of a foot) when it contains $1200 \mathrm { ft } ^ { 3 }$ of grain?
A) $6.99 \mathrm { ft }$
B) $7.88 \mathrm { ft }$
C) $6.28 \mathrm { ft }$
D) $7.5 \mathrm { ft }$
E) none of these

Question

Grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height. How high is the pile (to the nearest hundredth of a foot) when it contains $1200 \mathrm { ft } ^ { 3 }$ of grain? 
A) $6.99 \mathrm { ft }$
B) $7.88 \mathrm { ft }$
C) $6.28 \mathrm { ft }$
D) $7.5 \mathrm { ft }$
E) none of these

Quizplus · Accepted Answer

The volume $V$ of a cone is given by the formula $V = \frac{1}{3} \pi r^2 h$, where $r$ is the radius and $h$ is the height of the cone. Given that the diameter is always three times the height, we have $d = 3h$, which means $r = \frac{3h}{2}$. Substituting $r$ in the volume formula gives $V = \frac{1}{3} \pi \left(\frac{3h}{2}\right)^2 h = \frac{1}{3} \pi \frac{9h^2}{4} h = \frac{9}{12} \pi h^3 = \frac{3}{4} \pi h^3$.We are given that $V = 1200 \, \text{ft}^3$, so we can set up the equation $\frac{3}{4} \pi h^3 = 1200$. Solving for $h$, we get:$$h^3 = \frac{1200 \times 4}{3 \pi}$$$$h^3 = \frac{4800}{3 \pi}$$$$h^3 = \frac{1600}{\pi}$$$$h = \left(\frac{1600}{\pi}\right)^{\frac{1}{3}}$$$$h \approx 8.610 \, \text{ft}$$ (to the nearest hundredth of a foot).Since none of the provided options matches this calculation, the correct answer is E) none of these.

Grain Is Falling from a Chute onto the Ground, Forming 1200ft31200 \mathrm { ft } ^ { 3 }1200ft3

Grain Is Falling from a Chute onto the Ground, Forming $1200 \mathrm { ft } ^ { 3 }$