Solve the problem.
-A conical tank is resting on its apex. The height of the tank is $16 \mathrm { ft }$, and the radius of its top is $4 \mathrm { ft }$. The tank is full of gasoline weighing $45 \mathrm { lb } / \mathrm { ft } ^ { 3 }$. How much work will it take to pump the gasoline to a level $16 \mathrm { ft }$ above the cone's top? Give your answer to the nearest $\mathrm { ft }$ ' $\mathrm { lb }$.
A) 96,510 ft · lb
B) 241,274 ft · lb
C) 450,379 ft · lb
D) 48,255 ft · lb

Question

Quizplus · Accepted Answer

To find the work required to pump the gasoline, we use the formula for work, $W = \rho g h A d$, where $\rho$ is the density of the fluid, $g$ is the acceleration due to gravity (which is often omitted in problems like this since we're given weight density directly, making $g = 1$ for simplification), $h$ is the height the fluid is lifted, $A$ is the cross-sectional area at a given height, and $d$ is the differential height element. For a conical tank, the volume element $dV$ at a height $y$ from the apex is given by the formula for the volume of a cylinder, $dV = \pi r^2 dy$, where $r$ is the radius of the cross-section at height $y$. The radius $r$ at any height $y$ can be found using similar triangles, $r/y = R/H$, where $R = 4 \, \text{ft}$ is the radius of the base and $H = 16 \, \text{ft}$ is the total height of the cone. Thus, $r = (y/16) \cdot 4$. The work done to move a volume element $dV$ to the top of the tank is $dW = \rho g h dV$, where $h$ is the distance the element needs to be lifted to get to a point $16 \, \text{ft}$ above the top of the cone, so $h = 16 - y + 16 = 32 - y$. Substituting $dV$ and integrating from $y = 0$ to $y = 16$ gives the total work. The correct calculation, considering the weight density of gasoline ($45 \, \text{lb/ft}^3$) and the geometry of the problem, results in $241,274 \, \text{ft} \cdot \text{lb}$, making choice B the correct answer.

Solve the Problem 16ft16 \mathrm { ft }16ft , and the Radius of Its Top Is

Solve the Problem $16 \mathrm { ft }$ , and the Radius of Its Top Is