il spilling from a ruptured tanker spreads in a circular shape on the surface of the ocean.The area of the spill increases at a rate of 9π m2~min. How fast is the radius of the spill increasingwhen the radius is 10 m?A. 910 m/minB. 920 m/minC. 9π20 m/minD. 920π m/minE. 9π10 m/min
Question
il spilling from a ruptured tanker spreads in a circular shape on the surface of the ocean.The area of the spill increases at a rate of 9π m2~min. How fast is the radius of the spill increasingwhen the radius is 10 m?A. 910 m/minB. 920 m/minC. 9π20 m/minD. 920π m/minE. 9π10 m/min
Solution
The area of a circle is given by the formula A = πr², where r is the radius of the circle.
Given that the area of the oil spill is increasing at a rate of 9π m²/min, we can write this as dA/dt = 9π.
We want to find how fast the radius of the spill is increasing when the radius is 10 m, i.e., we want to find dr/dt when r = 10.
Differentiating both sides of the area formula with respect to time t, we get dA/dt = 2πr(dr/dt).
Substituting the given values into this equation, we get 9π = 2π10(dr/dt).
Solving for dr/dt, we get dr/dt = 9π / (20π) = 9/20 m/min.
So, the radius of the oil spill is increasing at a rate of 9/20 m/min when the radius is 10 m.
Therefore, the correct answer is C. 9π/20 m/min.
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