Math& $151-$Upsal

Daily Exercises $24$: More related rates practice $($Section $4\mathrm{.}1)$

$\backslash$table$[[$Learning objectives$],[$Topic$,$Importance for future classes,Where is it used in the future?$],[$Set up and solve related rates problems,High,$\backslash$table$[[$Science$,$ very common in$],[$engineering$]]]]$

You must complete two out of three of these for full credit on this assignment. If you complete all $3,$ you can earn $1$ bonus point. It is highly recommended to complete all $3!$

Exercise $1$: Relating surface area to volume

The surface area A and volume $V$ of a sphere are related by the following formula:

Suppose $V$ is increasing at a rate of $3$ cubic centimeters per second. How fast is A increasing when $V=10$ cubic centimeters? $($Hint: Use the original equation to find the value of A when $V=10)$

$\frac{\text{d}\text{A}\text{A}(t)}{dt}=\frac{dA}{dV}\frac{dV}{dt},\frac{dV}{dt}=3$

$A^3=\frac{4}{83}(10{)}^{2}3$

$3A^2=\frac{400}{3}$

Exercise $2$: Walking towards a tree

A man is walking toward a tree which is $10$ meters tall. Let $$ denote the angle of elevation from his line of sight to the top of the tree. Let $x$ denote the man's distance from the tree in meters.

a$)$ Find a formula which relates $$ and $x.$

$($Problem continued on next page$)$b$)$ Suppose that the man is walking at a rate of $1\mathrm{.}5$ meters per second. How fast is the angle of elevation changing when he is $4$ meters from the tree?