Calculus Help
I need step-by-step solutions please. Follow instructions on the sheets provided. Thank you.
Name: Related Rates Section:
4.1 Related Rates
1. A spherical balloon is being filled with air at a constant rate of 2 cm3/sec (see Example 4.1 on pg. 342
of the text). How fast is its radius increasing when the radius is 3 cm?
2. A rocket is launched so that it rises vertically (see example 4.3 on pg. 345 of the text). A camera is
positioned 5000 ft from the launch pad. When the rocket is 1000 ft above the launch pad, its velocity
is 600 ft/s. Find the necessary rate of change of the camera’s angle at this time so that it stays focused
on the rocket.
3. A 10-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/s,
how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 44
Name: Related Rates Section:
4. Two buses are driving along parallel freeways 5 mi apart, one heading east and the other heading west.
Assuming each bus drives a constant 55 mph, find the rate at which the distance between the buses is
changing when they are 13 mi apart.
5. A 6-ft tall person is walking away from a 10-ft lamppost at a constant rate of 3ft/s. What is the rate at
which the length of the person’s shadow is changing when the person is 10-ft from the pole.
6. A water trough in the shape of an isosceles trapezoidal prism (shown below) is being fill with water at
a constant rate of 32 cm
3
s . Determine the rate of change of the height when the height is 0.25 m.
0.6 m
1.2 m
3 m
0.5 m
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 45
Name: Related Rates Practice Section:
A.8 Related Rates Practice
1. Water flows onto a flat surface at a rate of 5 cm3/s forming a puddle that is 10 mm deep. How fast is
the radius growing when the radius is:
(a) 1 cm?
(b) 10 cm?
(c) 100cm?
2. A spherical balloon is inflated with air flowing at a rate of 10 cm3/s. How fast is the radius of the
balloon increasing when the radius is 6 cm?
3. An F-22 aircraft is flying at 500 mph with an elevation of 10,000 ft on a straight-line path that will
take it directly over an anti-aircraft gun. At what rate must the angle of elevation of the anti-aircraft
gun θ change in order to track the plane when it is 1 mile away. (See #7 on pg. 179 for help draw a
diagram.)
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 103
Name: Related Rates Practice Section:
4. A boat is being pulled toward a dock at a constant rate of 30 ft/min by a winch located 10 ft above the
deck of the boat (the point where the rope/cable from the winch is connected). Draw a diagram of the
situation, then determine the rate at which the boat is approaching the dock when it is 15 feet out.
5. An cylindrical cone, 20 ft deep and 10 ft across at the top, is being filled with water at a rate of 10
ft3/min. Determine the rate the water is rising in the tank when it is (a) 19 feet deep and (b) 10 feet
deep.
10 ft︷ ︸︸ ︷
20 ft
6. A 4 m ladder is leaning up against a wall. The base of the ladder is sliding away from the wall at a rate
of 3 cm/s. Determine the rate of change of the height dhdt , up to which the ladder reaches.
4
︷
︸︸
︷
h
3 cm/s →
For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1 104
Name Math 109 – Quiz 7 Section
1. Consider the equation xy2 − x2y = 6
(a) Use implicit differentiation to determine dyd x
(b) Determine the instantaneous rate of change dyd x of the equation at the point (2,−1)
(c) Find the equation of the line tangent to the graph of the equation at the point (2,−1)
Spring 2021 1
Name Math 109 – Quiz 7 Section
2. A spherical balloon is being filled with air (volume) at the constant rate of 2 cm3/sec. How fast is the radius
increasing when the radius is 3 cm? (Note: For a sphere, V = 43 πr
3)
Spring 2021 2