Jump to content
Sign in to follow this  

Rates of Change - Calculus Homework

Recommended Posts

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.

Share this post


Link to post
Share on other sites

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

Share this post


Link to post
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Reply to this topic...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Sign in to follow this  

×
×
  • Create New...