129k views
5 votes
A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 43 cm/s. Find the rate which the area within the circle is increasing after

a) 1 second, b) 3 seconds, and c) 5 seconds.
What can you conclude?

1 Answer

3 votes

The rate at which the area within the circle is increasing, obtained using the chain rule of differentiation, which indicates a linear relation with the duration of travel are;

a) 11617.61 cm²/s

b) 34852.83 cm²/s

c) 58088.05 cm²/s

There is a linear relationship between the rate of change of the area and the duration of travel of the ripple

What is a linear relation?

A linear relation is one in which the graph of the equation representing the relation is a straight line.

The chain rule of differentiation indicates that we get;

dA/dt = dA/dr × dr/dt

The area of a circle is; A = π·r²

dA/dr = 2·π·r

dr/dt = 43 cm/s

Therefore; dA/dt = 2·π·r × 43

dA/dt = 86·π·r

a) The radius of the ripple after 1 second is; r = 1 s × 43 cm/s = 43 cm

The radius after 1 second is r = 43 cm

The rate at which the area is increasing after 1 second is; 86 × π × 43 ≈ 11617.61 cm²/s

b) The radius, r, after 3 seconds is; 3 × 43 = 129 cm

The rate at which the area is increasing is; 86 × π × 129 ≈ 34852.83 cm²/s

c) The radius, r, after 5 seconds is; 5 × 43 = 215 cm

The rate at which the area is increasing is; 86 × π × 215 ≈ 58088.05 cm²/s

The first difference in the rate of change of the areas are;

34852.83 - 11617.61 = 23235.22

58088.05 - 34852.83 = 23235.22

The first rate of change in the area indicates a linear relationship, therefore, the area increases linearly with duration of travel of the circular ripple

User Darklord
by
7.5k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories