Related rates: When t is given and r is not

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Homework Help Overview

The problem involves related rates in the context of a circular ripple expanding in a pond. The radius of the ripple increases at a constant rate, and the task is to determine how quickly the area of the circle is increasing after a specific time has elapsed.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to understand how to incorporate time into their calculations, specifically questioning how to find the radius at 10 seconds. Some participants suggest starting with the area formula and connecting it to the rate of change of the radius.

Discussion Status

Participants are exploring the relationship between the radius and area of the circle, with some guidance provided on how to derive the radius at a given time. There is an acknowledgment of the challenges faced in the problem, but no consensus has been reached on a complete solution.

Contextual Notes

The original poster expresses frustration with the complexity of the problems in their take-home quiz, indicating that they often require additional steps to find necessary information.

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Homework Statement


A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 3 ft/s. How rapidly is the area enclosed by the ripple increasing at the end of 10 s?

Homework Equations


How do I even set up implicit differentiation?

The Attempt at a Solution


I know that dr/dt is 3 ft/s and that I'm looking for dA/dt, but I don't know where t = 10s goes into that equation, especially when I don't have the radius... Do I multiply time times dr/dt to get radius at 10 seconds and go from there?

HELP!
 
Last edited:
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Start with the formula for the area of a circle. You are given how the radius changes with time. Do you see how those might fit together to get an expression for the change in area? Start there and see if that helps you out.

Edit: Yes, you can get the radius at 10 seconds that way.
 
well, i know that dA/dt = 2pi*radius*dr/dt, but I don't have the radius...
 
At 10 seconds you do. Actually at any second you do.
 
Sweetness... Thanks so much! I've been going back and forth thinking that was just too easy. All the problems on this take home quiz have that one extra step you have to go thru to get all the info. It's making me crazy!
 
becca4 said:
It's making me crazy!

Yeah, calculus can do that to you. LOL
 

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