Water leaking onto a floor creates a circular pool with an area that increases at the rate of 3 square inches per minute. How fast is the radius of the pool increasing when the radius is 10 inches?
You have to be kidding!! The entire area is only growing at 3 in^2/s.
You need to learn to do a sanity check on any answer you get, especially in a new situation.
Using the given equation and numbers, we have
3 = 2π*10 dr/dt
dr/dt = 3/(20π) = 0.0477 in/s
Your answer is 60π. You plugged in 3 for dr/dt, but that was incorrect.
well, a = πr^2
da/dt = 2πr dr/dt
...
Well, well, well, looks like we have some leaky business here! Let's see what we can do. We know that the area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
Now, we are given that the area is increasing at a rate of 3 square inches per minute, which means dA/dt = 3 in^2/min. We want to find how fast the radius is increasing, which is dr/dt.
To solve the problem, we'll need to use the chain rule. Differentiating both sides of the equation A = πr^2 with respect to time, we get dA/dt = 2πr(dr/dt).
Now, we know dA/dt = 3 in^2/min, and we want to find dr/dt when the radius is 10 inches. Plugging these values into our equation, we get:
3 = 2π(10)(dr/dt)
Now, it's just a matter of solving for dr/dt. Dividing both sides of the equation by 20π, we find:
dr/dt = 3/(20π) ≈ 0.0477 inches per minute.
So, the radius of the pool is increasing at approximately 0.0477 inches per minute when the radius is 10 inches. Stay dry, my friend!
To find how fast the radius of the pool is increasing, we can use the formula for the area of a circle:
A = πr²,
where A is the area and r is the radius.
We are given that the area is increasing at a rate of 3 square inches per minute, so we can write this as:
dA/dt = 3 square inches per minute.
To find how fast the radius is increasing, we need to find dr/dt, the rate at which the radius is changing with respect to time.
To do this, we can take the derivative of the area equation with respect to time:
dA/dt = d(πr²)/dt.
Since π is a constant, we can take it out of the derivative:
dA/dt = π * d(r²)/dt.
To simplify further, we can use the power rule of differentiation:
d(r²)/dt = 2r * dr/dt.
Substituting this back into the equation, we have:
dA/dt = π * 2r * dr/dt.
Now we can solve for dr/dt:
dr/dt = (dA/dt) / (π * 2r).
Substituting the given value of dA/dt = 3 square inches per minute and r = 10 inches, we can calculate the answer:
dr/dt = (3) / (π * 2 * 10).
Simplifying this expression, we have:
dr/dt = 3 / (20π).
Therefore, when the radius is 10 inches, the rate at which the radius is increasing is 3 / (20π) inches per minute.