Homework

Problem 1 · easy

A circle's radius grows at 2 cm/s. How fast is the area growing when r = 4 cm?

Problem 2 · easy

A circle's radius shrinks at 1 cm/s. How fast is the circumference changing?

Problem 3 · easy

A sphere's radius grows at 3 cm/s. How fast is the volume growing when r = 2 cm?

Problem 4 · medium

A balloon is inflated so V = (4/3)*pi*r^3 grows at 50 cm^3/s. Find dr/dt when r = 5.

Problem 5 · medium

A 13-ft ladder leans on a wall. The base slides at 2 ft/s. How fast does the top slide down when base is 5 ft from wall?

Problem 6 · medium

Both cars leave an intersection, one north at 60 mph, one east at 80 mph. How fast are they separating after 1 hour?

Problem 7 · medium

A water tank is a cylinder with radius 3 m. Water drains at 9 m^3/min. How fast is the water level dropping?

Problem 8 · challenge

A 6-m pole casts a shadow. The sun lowers at a rate that makes the shadow lengthen at 0.5 m/s. At what rate is the angle of elevation of the sun changing when the shadow is 6 m long? (Leave in terms of trigonometry if needed, or use the relation tan(theta)=6/s.)

Problem 9 · challenge

A conical tank (radius 4 m, height 8 m) drains at 4 m^3/min. How fast does the water level drop when h = 4 m?

Problem 10 · challenge

A person 1.8 m tall walks at 1 m/s toward a 9 m light. How fast is their shadow shrinking (getting shorter)?

Problem 11 · medium

A circle's area grows at 6*pi cm^2/s. How fast is the radius growing when r = 3?

Problem 12 · medium

A sphere's surface area is A = 4*pi*r^2. If the radius grows at 0.5 cm/s, how fast is A growing when r = 10?

Problem 13 · easy

x and y satisfy x^2 + y^2 = 25. If dx/dt = 3 when x = 4, y = 3, find dy/dt.

Problem 14 · medium

x and y satisfy x^2 + y^2 = 169. If x = 5, y = 12, and dx/dt = -2, find dy/dt.

Problem 15 · easy

A box's length is fixed at 4, width is fixed at 3. Height grows at 2 in/min. How fast is the volume growing?

Problem 16 · easy

A cylinder (radius 5, variable height) is filled at 25*pi cm^3/s. How fast does the height rise?

Problem 17 · easy

A right triangle has legs x and y with x + y = 20 (constant perimeter base). dx/dt = 3. Find dy/dt.

Problem 18 · medium

The area of a right triangle is A = (1/2)*x*y. If x = 5, y = 4, dx/dt = 2, dy/dt = -1, find dA/dt.

Problem 19 · medium

A 10 m ladder leans on a wall. If the top slides down at 0.5 m/s and the top is at y = 6, find dx/dt (speed of base sliding out).

Problem 20 · medium

A circular oil spill grows at 3*pi m^2/s. How fast is the radius growing when A = 25*pi m^2?

Problem 21 · medium

A gas balloon has V = (4/3)*pi*r^3. Gas leaks at 8*pi cm^3/s. Find dr/dt when r = 2.

Problem 22 · medium

Two boats leave a dock. Boat A goes east at 10 knots, Boat B goes north at 24 knots. How fast are they separating after 1 hour?

Problem 23 · medium

A ladder 5 m long leans on a wall. When the base is 3 m from the wall and sliding out at 1 m/s, how fast is the top dropping?

Problem 24 · easy

A cube's side length grows at 2 in/s. How fast is the volume growing when side = 5 in?

Problem 25 · medium

A cube's side shrinks at 0.5 cm/s. How fast is the surface area decreasing when side = 10 cm?