Solve the problem.
-A rectangular swimming pool has a parabolic drain plate at the bottom of the pool. The drain plate is shaped like the region between $y = \frac { 1 } { 2 } x ^ { 2 }$ and the line $y = \frac { 1 } { 2 }$ from $x = - 1$ to $x = 1$. The pool is $10 \mathrm { ft }$ by $20 \mathrm { ft }$ and $8 \mathrm { ft }$ deep. If the drain plate is designed to withstand a fluid force of $200 \mathrm { lb }$, how high can the pool be filled without exceeding this limitation?
A) 5 ft 1 in.
B) 5 ft 6 in.
C) 5 ft 9 in.
D) 4 ft 11 in.

Question

Quizplus · Accepted Answer

The fluid force on the drain plate is calculated by integrating the pressure over the area of the plate. The pressure at depth $h$ is $\rho gh$, where $\rho$ is the density of water and $g$ is the acceleration due to gravity. Solving the integral for the given parabolic region and setting it equal to 200 lb gives the maximum height $h$ the pool can be filled. This calculation results in approximately 5 ft 1 in.

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