Water leaking onto a floor forms a circular pool. The area of the pool increases at a rate of 16 picm^(2)//min. How fast is the radius of the pool increasing when the radius is 7cm ?

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Water leaking onto a floor forms a circular pool. The area of the pool increases at a rate of  16 picm^(2)//min. How fast is the radius of the pool increasing when the radius is  7cm ?

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Area Formula: The area A of a circle is given by the formula A = πr^2, where r is the radius of the circle. We are given that the area of the pool increases at a rate of 16 picm^2/min, which is dA/dt = 16 picm^2/min. We need to find the rate at which the radius r is increasing, which is dr/dt, when the radius is 7cm.Chain Rule Application: To find dr/dt, we will use the chain rule from calculus, which relates the rates of change of two related quantities. Differentiating both sides of the area formula with respect to time t, we get dA/dt = d(πr^2)/dt. Applying the chain rule, we get dA/dt = 2πr * dr/dt.Substitution and Calculation: We can now solve for dr/dt by substituting the given rate of change of the area and the radius at the moment we are interested in. We have dA/dt = 16 picm^2/min and r = 7cm. Plugging these values into the equation, we get 16 = 2π * 7 * dr/dt.Isolating dr/dt: Solving for dr/dt, we divide both sides of the equation by 2π * 7 to isolate dr/dt. This gives us dr/dt = 16 / (2π * 7).Final Calculation: Now we perform the calculation: dr/dt = 16 / (2 * π * 7) = 16 / (14π) = 8 / (7π) cm/min.

Water leaking onto a floor forms a circular pool. The area of the pool increases at a rate of $16\,\text{picm}^{2}/\text{min}$ . How fast is the radius of the pool increasing when the radius is $7\,\text{cm}$ ?

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Water leaking onto a floor forms a circular pool. The area of the pool increases at a rate of 16 picm2/min16\,\text{picm}^{2}/\text{min}16picm2/min. How fast is the radius of the pool increasing when the radius is 7 cm7\,\text{cm}7cm?

Water leaking onto a floor forms a circular pool. The area of the pool increases at a rate of $16\,\text{picm}^{2}/\text{min}$ . How fast is the radius of the pool increasing when the radius is $7\,\text{cm}$ ?