The area A of a circle is increasing at a constant rate of 1.5 cm2s-1. Find, t

Question

The area A of a circle is increasing at a constant rate of 1.5 cm²s^-1. Find, to 3 significant figures, the rate at which the radius r of the circle is increasing when the area of the circle is 2 cm².

Answer 1

The area A of a circle is increasing at a constant rate of 1.5 cm²s^-1. Find, to 3 significant figures, the rate at which the radius r of the circle is increasing when the area of the circle is 2 cm².

0.300 cms⁻¹
0.299 cms⁻¹ ✓
0.798 cms⁻¹
0.200 cms⁻¹

Explanation

Step 1: Write area formula and differentiate
A = πr²
Differentiate with respect to time: dA/dt = 2πr × dr/dt

Step 2: Identify known values
dA/dt = 1.5 cm²/s (rate of area change)
A = 2 cm² (current area)
Need to find: dr/dt

Step 3: Find radius when A = 2
πr² = 2
r² = 2/π
r = √(2/π)
r ≈ 0.7979 cm

Step 4: Substitute into differentiated equation
1.5 = 2π(0.7979) × dr/dt
1.5 = 5.0133 × dr/dt

Step 5: Solve for dr/dt
dr/dt = 1.5/5.0133
dr/dt ≈ 0.2992
dr/dt ≈ 0.299 cm/s (to 3 sig figs)