The angle of a sector of a circle is 75 If the radius of this circle is increasing at the rate of 0.3cm/s find the rate of increase of the area when the radius is 6cm

a = 75/360 * π * r^2

da/dt = 75/360 * π * 2r dr/dt

da/dt = 75/360 * π * 2 * 6 * 0.3

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To find the rate of increase of the area when the radius is 6 cm, we first need to calculate the initial area of the sector.

The area of a sector can be calculated using the formula:

Area = (θ/360) × π × r^2

where θ is the angle of the sector in degrees, π is a mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

In this case, the angle of the sector is given as 75 degrees, and the radius is initially 6 cm. Therefore, the initial area can be calculated as:

Area = (75/360) × π × (6^2) = (5/24) × π × 36 ≈ 23.56 cm^2

Now, to find the rate of increase of the area, we can differentiate the formula for the area of a sector with respect to time. The rate of change of the area (dA/dt) can be expressed as:

dA/dt = (θ/360) × π × (2r × dr/dt)

Here, dr/dt represents the rate of change of the radius with respect to time. We are given that the radius is increasing at a rate of 0.3 cm/s, so dr/dt = 0.3 cm/s.

Plugging in the values into the formula, we get:

dA/dt = (75/360) × π × (2 × 6 × 0.3) = (5/24) × π × 3.6 ≈ 2.94 cm^2/s

Therefore, the rate of increase of the area when the radius is 6 cm is approximately 2.94 cm^2/s.