Note: Be careful of units, and be sure to look up conversion factors, as needed.

A water tank is made in the shape of an ellipse, with height $30$ feet and width $25$ feet, rotated about its central axis. Recall that the equation of an ellipse centered at the origin is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $a$ is the radius in the $x$ direction and $b$ is the radius in the $y$ direction. Assuming the tank is half full, find the work done to pump all of the water out over the top of the tank and into a tanker truck, $10$ feet above the tank.

$($a$)$ Explain what is wrong with the following solution to the problem.

Work $=($ Force $)*($ Distance $)=($ Total weight $of$ water $)**(15ft)$

$($b$)$ With the origin at the center and the $y-$axis running vertically through the center of the tank, consider one "slice" of water at level $y$ and with thickness $dy.$ Draw the slice and calculate the amount of work done to lift this "slice" of water up to the tanker$.$

$($c$)$ Set up and evaluate an integral to calculate the total amount of work done to pump all of the water over the top of the tank.