astonky Posted November 1, 2017 Report Share Posted November 1, 2017 Hi everyone, I'm struggling with my calculus homework on the rates of change. I have a number of problems I need to do, but I don't understand at all how to solve them. For the example below, could someone please walk me through the exact steps (and why you're doing them)? A spark from a fire burns a hole in a paper napkin. The hole initially has a radius of 1cm and its area is increasing at a rate of 2cm²/s. Find the rate of change of the radius when the radius is 5 cm. Thank you very much in advance for helping me out. Reply Link to post Share on other sites More sharing options...
Nomenclature Posted November 2, 2017 Report Share Posted November 2, 2017 First, let's write down the equation for the area of a circle: A = (pi)r2 The question asks for the rate of change of the radius. To find this, we must take the derivative of the area function with respect to time. This is because the rate of change of the area can be obtained from the rate of change of the radius and vice versa. Thus, we differentiate both sides (d/dt) (da/dt) = (d/dt) (pi)r2 (da/dt) = (pi) 2r (dr/dt) note: pi is a constant; the chain rule is applied. 2r = (dr/da) and we multiply that by (dr/dt) (we don't actually know the function for t in terms of r) We know the value for (da/dt), it's just how much the area is changing with respect to time, which was given to us: 2 cm2/s. 2 = (pi) 2r (dr/dt) Likewise, r was given to us as 5 2 = (pi) (2*5) (dr/dt) 2 = 10pi (dr/dt) (2/(10pi)) = (dr/dt) (dr/dt) = (1/5)pi 1 Reply Link to post Share on other sites More sharing options...
astonky Posted January 25, 2018 Author Report Share Posted January 25, 2018 I know my response is kinda late, but thankks very much. Your explanation really helped; I just forgot to reply in november. Reply Link to post Share on other sites More sharing options...
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