Question

3. A conical water tower has a height of 12ft and a radius of 3ft. Water is pumped into the tank at a rate of 4ft³ /min.

a. At what rate is the water level changing at the time t ₀ when the water level is 6ft ?
b.At what rate is the radius of the top of the water in the tank changing at the time t ₀ when the water level is 6ft ? 4. A ship is 40 miles west of a lighthouse. The ship is heading north at a rate such that the angle θ(t) is changing at a constant rate of π/18 radians per hour. At what rate is the distance, x(t), between the ship and the lighthouse changing at the time t₀ when θ(t₀) )=π/6 radians?

Answer 1

The student's questions involve applying related rates in calculus to a conical water tower problem and trigonometric methods to determine the rate of change of distance between a ship and a lighthouse.

Step-by-step explanation:

The two questions mentioned are related to different applications of calculus and physics principles.

Conical Water Tower

For the conical water tower with a height of 12ft and a radius of 3ft being filled at a rate of 4ft3/min, we can apply the concept of related rates to find the rate at which the water level is changing and the rate at which the radius of the top of the water is changing when the water level is 6ft.

Ship and Lighthouse Problem

In the ship and lighthouse problem, we are given that a ship is 40 miles west of a lighthouse and heads north in such a way that the angle θ(t) is changing at a constant rate. Using trigonometric relationships, we can find the rate of change of the distance x(t) between the ship and the lighthouse when θ(t0) = π/6 radians.