Answer:
Step-by-step explanation:
The question is incomplete. Here is the complete question.
Find the average rate of change of the area of a circle with respect to its radius r as r changes from 3 to each of the following.
(i) 3 to 4
(ii) 3 to 3.5
(iii) 3 to 3.1
(b) Find the instantaneous rate of change when r = 3. A'(3)
Area of a circle A(r)= πr²
The average rate of change of the area of a circle with respect to its radius
ΔA(r)/Δr = πr₂²-πr₁²/r₂-r₁
ΔA(r)/Δr = π(r₂²-r₁²)/r₂-r₁
i) If the radius changes from 3 to 4
ΔA(r)/Δr = π(r₂²-r₁²)/r₂-r₁
ΔA(r)/Δr = π(4²-3²)/4-3
ΔA(r)/Δr = π(16-9)/1
ΔA(r)/Δr = 7π
<em>Hence, average rate of the area of a circle when the radius changes from 3 to 4 is 7π</em>
ii) If the radius changes from 3 to 3.1
ΔA(r)/Δr = π(r₂²-r₁²)/r₂-r₁
ΔA(r)/Δr = π(3.5²-3²)/3.5-3
ΔA(r)/Δr = π(12.25-9)/0.5
ΔA(r)/Δr = 3.25π/0.5
ΔA(r)/Δr = 6.5π
<em>Hence, average rate of the area of a circle when the radius changes from 3 to 3.5 is </em>6.5π
iii) If the radius changes from 3 to 3.1
ΔA(r)/Δr = π(r₂²-r₁²)/r₂-r₁
ΔA(r)/Δr = π(3.1²-3²)/3.1-3
ΔA(r)/Δr = π(9.61-9)/0.1
ΔA(r)/Δr = 0.61π/0.1
ΔA(r)/Δr = 6.1π
<em>Hence, average rate of the area of a circle when the radius changes from 3 to 3.1 is </em> 6.1π
iv) Instantaneous rate of change A'(r) = 2πr
When r = 3;
A'(3) = 2π(3)
A'(3) = 6π
<em>Hence, the instantaneous rate of change when r = 3 is 6π </em>
