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