Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 pounds/sq.in., the laughing gas chamber inside the rocket will explode. Tiffany worked from a formula $$p = (14.4)e^{-\frac{h}{10}}$$ pounds/sq.in. for the atmospheric pressure $h$ miles above sea level. Assume that the rocket is launched at an angle of α above level ground at sea level with an initial speed of 1400 feet/sec. Also, assume the height (in feet) of the rocket at time t seconds is given by the equation $y(t) = −16(t^2) + 1400tsin(α)$.
If the angle of launch is $α = 14°$, determine the minimum atmospheric pressure exerted on the rocket during its flight.
So I know I have to plug in the 14 for a but don't I need t as well to get the y(t) which is the height to find the pressure?
Note: Mathematically, the "correct" input for sine is in radians. For ease of notation, assume degrees below.
You want to find maximal height by optimising $y(t)$ and then plugging in $h$. Note that $y'(t) = -32t + 1400\sin(14) = 0$, whence $t = 700\sin(14)/16$. So, $h = (1400\sin(14))^2\cdot(1/64)$. Then solve $p(h) = 14.4\cdot \exp(-h/52800)$, where we convert from feet to miles.