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$\begin{minipage}{1.09\textwidth}\large\texttt{\#3.12} \textsf{A sand-spraying locomotive sprays sand horizontally into a freight car as shown in the sketch. The locomotive and freight car are not attached. The engineer in the locomotive maintains his speed so that the~distance to the freight car is constant. The sand is transferred at a~rate \begin{math}dm/dt\end{math} = 10 kg/s with a velocity 5 m/s relative to the locomotive. The car starts from rest with an initial mass of 2,000 kg. Find its speed after 100 s.}\end{minipage}$

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$\frac{\partial ^2 \left( \ln {\tan x}\right)}{\partial x^2}=\sec^2x-\csc ^2 x \leq 0 \ \ \forall x\in[0,\frac{\pi}{4}] \\ \\\texttt{hence the function is concave }\\ \\ \frac{\ln {\tan A}+\ln {\tan B}}{2} \leq \ln {\tan \frac{(A+B)}{2}} \\ \ln(\tan A.\tan B)\leq \ln \frac{1}{3} \\ \tan A.\tan B \leq \frac{1}{3}$

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$\begin{minipage}{1.10\textwidth}\large\texttt{\#2.37} \textsf{The Eureka Hovercraft Corporation wanted to hold hovercraft~races as an advertising stunt. \, The hovercraft supports itself by blowing air downward, \, and has a big fixed propeller on the top deck for forward propulsion. \,Unfortunately, \,it has no steering equipment, \,so that the pilots found that making high speed turns was very difficult. The company decided to overcome this problem by designing a bowl shaped track in which the hovercraft, once up to speed, would coast along in a circular path with no need to steer. They hired an engineer to design and build the track, and when he finished, he hastily left the country. When the company held their first race, they found to their dismay that the craft took exactly the same time \begin{math}T\end{math} to circle the track, no matter what its speed. Find the equation for the cross section of the bowl in terms of \begin{math}T\end{math}.}\end{minipage}$