Question

a) A full conical water tank has height 3 m and diameter across the top 2 m. It is leaking water at a rate of 10,000 cm^3 per minute. How fast is the height of the water decreasing when the volume of water is 3,000,000 cm^3 ?

b) Two planes are approaching the same air traffic control tower at equal and constant altitude. Plane A is heading due north at 200 mph, while Plane B is heading due west at 175 mph. At what rate is the distance between the two planes closing when plane A is 60 mi away and plane B is 120 mi away?

c) A spherical balloon has radius 75 cm. Use differentials to estimate the volume of rubber given that the rubber is 1 mm thick.

d) Find the absolute maximum and minimum values of f(x) = 2x^3 − 3x^2 − 12x + 1 on the closed interval [−2, 3].

e) Evaluate lim t→0 (e^2t − 1) / sin(t)

Thank you!

Answer 1

a

Solution-

The volume of the cone is given as,

Leaking rate = 10000 cm^3/min

The volume in the tank at any given time T is given as,

use a similar triangle,

h when the volume is 3000000 cm^3,

h=295.5 cm

differentiate,

rate of height decreasing is 0.3280 cm/min.