Intro to Calc

Intro to Calc Section 4.6

“ Related Rates”

1. Let A be the area of a square whose sides have length x, and assume that x varies with time. a. How are dA/dt and dx/dt related? b. At a certain instant the sides are 3 feet long and growing at a rate of 2 ft/min. How fast is the area growing at the instant?

2. Let V be the volume of a cylinder having height h and radius r, and assume that h and r vary with time. a. How are dV/dt, dh/dt, and dr/dt related? b. At a certain instant, the height is 6 inches and increasing at 1 in/sec, while the radius is 10 inches and decreasing at 1 in/sec. How fast is the volume changing at that instant? Is the volume increasing or decreasing at that instant?

3. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 6 mi2/hr. How fast is the radius of the spill increasing when the area is 9mi2?

4. A spherical balloon is to be deflated so that its radius decreases at a constant rate of 15 cm/min. At what rate must air be removed when the radius is 9 cm?

5. A 13-ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2 ft/sec, how fast will the foot be moving away from the wall when the top is 5 feet above the ground?

6. A boat is pulled into a dock by means of a rope attached to a pulley on the dock (see figure). The rope is attached to the bow of the boat at a point 10 feet below the pulley. If the rope is pulled through the pulley at a rate of 20 ft/min, at what rate will the boat be approaching the dock when 125 feet of rope is out?

xy 3 8 7. A particle is moving along the curve whose equation is  . Assume that the 1 y 2 5 x-coordinate is increasing at a rate of 6 units/sec when the particle is at point (1, 2). a. At what rate is the y-coordinate of the point changing at that instant? b. Is the particle rising or falling at that instant?

8. A rocket, rising vertically, is tracked by a radar station that is on the ground 5 miles from the launch pad. How fast is the rocket rising when it is 4 miles high and its distance from the radar station is increasing at a rate of 2000mi/hr? Answers: 1a. dA/ dt  2x  dx / dt 1b. dA/ dt  12 ft / min

2a. dV / dt  2rh  dr / dt  r 2  dh / dt 3 2b. dV / dt  20 in / sec

dA / dt 3. dr / dt  2r  dr / dt  mi / hr 

4. dV / dt  4r 2  dr / dt dV / dt  4860 cm3 / min

 y  dy / dt 5. dx / dt  x 5 dx / dt  ft / sec 6

z  dz / dt 6. dx / dt  x dx / dt  20.06 ft / min

5y 2  dx / dt 7. dy / dt  16 15xy 60 dy / dt   units / sec 7

z  dz / dt 8. dy / dt  y dy / dt  3201.56 mi / hr