Water is falling on a surface, wetting a circular area that is expanding at a rate of 10 mm²/sec.

How fast is the radius of the wetted area expanding when the radius is 116 mm? (Round approximations to four decimal places.)

A. What is the rate of expansion of the wetted area?

B. How fast is the radius increasing when it's 116 mm?

C. What is the area of the wetted surface when the radius is 116 mm?

D. Determine the time it takes for the radius to reach 116 mm.

This is a calculus problem addressing the concept of derivatives. The area of the circle expands at a rate of 10mm²/sec. The problem involves finding the rate at which the radius expands when it is 116mm by setting up an equation based on the derivative of the circle's area and solving for dr/dt.

Explanation:

This question refers to the application of the formula for the change in area of a circle with respect to time. To solve, we utilize the concept of derivatives in calculus, specifically applied to geometrical concepts.

Given that the area A of a circle is given by πr², the rate of change of the area (dA/dt) is 10mm²/sec. We want to find how rapidly the radius is changing (dr/dt) when r is 116mm. The derivative of the area with respect to time, dA/dt, can be expressed as 2πr(dr/dt). Setting dA/dt equal to 10mm²/sec gives us the equation: 10=2πr(dr/dt).

Substituting r=116 into the equation yields 10 = 2*pi*116*dr/dt. Solving this equation will give us the rate at which the radius is increasing when r=116mm (or, dr/dt).

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