Math 165 - Homework Assignment 3 Solution
Name:
Write your solutions to these problems on a separate sheet of paper. Show all work to re-
ceive full credit for each problem. Turn in complete, legible, organized, and logically sound
solutions and arguments. Give exact answers, not decimal approximations. This assignment
is worth 10 points and is due Tuesday, February 26 in class. I will grade all 3 problems.
1. Find ds
du if tan(su2)−3
√su + 1 = πu
Solution: ds
du =
π−2su sec2(su2) + s
2(su + 1)2/3
u2sec2(su2)−
u
3(su + 1)2/3
2. You are climbing a rigid 20-foot ladder that is leaning against a tall vertical wall when
the ladder begins to slide down the side of the wall. Assuming that the angle between
the ground and the ladder is decreasing at a consant rate of π
6radians per second, how
fast is the top of the ladder moving down the side of the wall when the top of the ladder
is 10 feet above the ground?
Solution: Referring the picture, we are given that dθ
dt =−π
6and we are trying to
find dh
dt . But sin(θ) = h
20 ⇒cos(θ)dθ
dt =1
20
dh
dt ⇒dh
dt = 20 cos(θ)dθ
dt
When h= 10, θ=π
6⇒dh
dt = 20 cos π
6 −π
6=−5π√3
3feet per second
3. The water in a full conical metal tank (with vertex pointing upward!) begins to drain out
at a rate of 4 cubic feet per minute. If the height of the tank is 18 feet and the diameter
of the bottom of the tank is 12 feet, determine how fast the water level is dropping when
the water is at a height of 10 feet.
Solution: Referring to the picture, we are given dV
dt =−4 and we are trying to find
dh
dt . The volume of water is given by V=1
3π(6)2(18) −1
3πr2h. But due to similar
right triangles, r
h=6
18 ⇒r=h
3.
Thus V= 36(6)π−π
27 h3⇒dV
dt =−πh2
9
dh
dt .
When the water level is 10 feet, h= 18 −10 = 8 ⇒dh
dt =9
16 πfeet per minute. Hence
the water level is dropping at a rate of −9
16 πfeet per minute.