I have an older Calculus book that I'm using to prepare for Calculus II. I'm just trying to brush up on some things. I find I often get thrown off by word problems.
"An object increases in height x above the surface of the earth, its weight w decreases because the pull of gravity on the object decreases. Suppose that a rocket weighs 80/(x + 4000)^2 million pounds at an altitude of x miles and is rising at the constant rate of 100 mi/min. Find the rate of change of weight with respect to time at any instant."
I'm given the answer dw/dt = -16,000/(x + 4000)^3, but I'm not sure how to arrive at this answer.
I suspect this has something to do with the Chain Rule. I know the derivative of (x + 4000)^2 = 2(x + 4000), gravity is s = 16t^2 and that the acceleration of the rocket is a=100. The velocity of the rocket would be v(t)=100t and the distance would be d(t) = 50t^2. Unfortunately I am not able to put everything together to solve this equation.
Best Answer
We have $w = \displaystyle \frac{80}{(x+4000)^2}$, so
$\displaystyle\frac{dw}{dt} = -\frac{160}{(x + 4000)^3}\frac{dx}{dt}$.
We are given that the rocket changes in height at a constant rate of $100$ miles per pinute. So thus $\frac{dx}{dt} = 100$.
Then $\displaystyle\frac{dw}{dt} = -\frac{160}{(x + 4000)^3}\frac{dx}{dt} = -\frac{16,000}{(x + 4000)^3}$, as desired.