Calculus Take-home assignment: Lessons 3.4-3.5 Name: ______________________________ Due: ______
Directions: For each related rate problem, show all 5 steps to solve. Circle your final, simplified answer.
Include units.
1. As a balloon in the shape of a sphere is being blown up, the volume is increasing at the rate of 4 cubic inches
per second. At what rate is the radius increasing when the radius is 1 inch?
2. A conical tank (with vertex down) is 10 feet across the top and 18 feet deep. As the water flows into the
tank, the change is the radius of the water is at a rate of 2 feet per minute, find the rate of change of the
volume of the water when the radius of the water is 2 feet.
3. A balloon rises at the rate of 8 feet per second from a point on the ground 60 feet from an observer. Find the
rate of change in the angle of elevation when the balloon is 25 feet above the ground.
4. A 5-meter-long ladder is leaning against the side of a house. The foot of the ladder is pulled away from the
house at a rate of 0.4 m/sec.
a. Determine how fast the top of the ladder is descending when the foot of the ladder is 3 meters from the
house.
b. Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which
the area of the triangle is changing when the foot of the ladder is 3 meters from the house.
c. Find the rate at which the angle between the ladder and the wall of the house is changing when the foot of
the ladder is 3 meters from the house.
5.. Find the local linear approximation of f(x) = 2x2 – 5x at the point where x = 1. Use your approximation to
estimate f(0.8), and f(1.2)
f(0.8) ≈ ______________
f(1.2) ≈ ______________
6. Find dy and Δy for f(x) = 3x2 + 2x at x = 2 and dx = Δx = .02.
dy =
Δy =
7. The measurement of the edge of a floor tile is found to be 12 inches, with a possible error of ±0.03 inch.
Use differentials to estimate the maximum possible error in the area of the floor tile. Find the percent
errors in the edge length and area of the floor tile.