91Ó°ÊÓ

What graphical feature must a function \(f\) possess for it to be its own inverse?

The National Weather Service uses the following formula to calculate the wind chill: $$ W=35.74+0.6215 T_{a}-35.75 V^{0.16}+0.4275 T_{a} V^{0.16} $$ where \(W\) is the wind chill temperature in \({ }^{\circ} \mathrm{F}, T_{a}\) is the air temperature in \({ }^{\circ} \mathrm{F},\) and \(V\) is the wind speed in miles per hour. Note that \(W\) is defined only for air temperatures at or lower than \(50^{\circ} \mathrm{F}\) and wind speeds above 3 miles per hour. (a) Suppose the air temperature is \(42^{\circ}\) and the wind speed is 7 miles per hour. Find the wind chill temperature. Round your answer to two decimal places. (b) Suppose the air temperature is \(37^{\circ} \mathrm{F}\) and the wind chill temperature is \(30^{\circ} \mathrm{F}\). Find the wind speed. Round your answer to two decimal places.

The period of a pendulum in seconds is given by $$ T=2 \pi \sqrt{\frac{L}{g}} $$ (for small displacements) where \(L\) is the length of the pendulum in meters and \(g=9.8\) meters per second per second is the acceleration due to gravity. My Seth-Thomas antique schoolhouse clock needs \(T=\frac{1}{2}\) second and I can adjust the length of the pendulum via a small dial on the bottom of the bob. At what length should I set the pendulum?