Math 180: Calculus I Fall 2014
October 23
TA: Brian Powers
1. The hypotenuse of an isosceles right triangle is decreasing in length at a rate of 4 m/s.
(a) At what rate is the area of the triangle changing when the legs are 5m long?
SOLUTION: First we should define the variables. Let xbe the length of the legs, and hthe
length of the hypotenuse.
Since the legs of the isosceles right triangle are in ratio of 1 : √2 to the hypotenuse, the rate of
change of the legs will be proportional, i.e. x=h/√2. Since dh
dt =−4,
dx
dt =dh
dt
1
√2=−4
√2=−2√2
Because the area A=x2/2,
dA
dt =dA
dx
dx
dt =x(−2√2 = −2√2xm/s
When x= 5, dA/dt =−2√25 = −10√2 m2/s.
(b) At what rate are the length of the legs of the triangle changing?
SOLUTION: The legs are decreasing at a constant rate of 2√2 m/s.
(c) At what rate is the area of the triangle changing when the area is 4 m2?
SOLUTION: The area is 4 m2when x2/2=4⇒x2= 8 so x= 2√2. At that time dA/dt =
−(2√2)2=−8 m2/s.
2. A swimming pool is 50m long and 20m wide. Its depth decreases linearly along the length from 3m to
1m. It is initially empty and filled with water at 1 m3/min.
(a) How fast is the water level rising 250 minutes after the filling begins?
SOLUTION: First of all, we need to talk about the dimensions of this. The inclined portion of
the pool has height 2m and length 50m. As a height his filled in, it will have a length bthat is
proportional to 50/2, so b= 25h.
16-1
