Related rates: When t is given and r is not

In summary, the problem involves a stone dropped into a pond and the resulting ripple that increases at a constant rate of 3 ft/s. The question asks for the rate of change of the area enclosed by the ripple after 10 seconds. To solve this, you can use the formula for the area of a circle and the given information about the change in radius with time. By setting up an expression for the change in area, you can find the radius at 10 seconds and solve for the desired rate of change.
  • #1
becca4
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0

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!
 
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  • #2
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.
 
  • #3
well, i know that dA/dt = 2pi*radius*dr/dt, but I don't have the radius...
 
  • #4
At 10 seconds you do. Actually at any second you do.
 
  • #5
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!
 
  • #6
becca4 said:
It's making me crazy!

Yeah, calculus can do that to you. LOL
 

FAQ: Related rates: When t is given and r is not

1) How do you find the rate of change when time is given and another variable is not?

To find the rate of change in this scenario, we use the chain rule. The formula is as follows:

dA/dt = (dr/dt) * (dA/dr)

Where A represents the quantity we are trying to find the rate of change for, t represents time, and r represents the other variable given. We can then plug in the values given for t and r to solve for the rate of change.

2) Can you give an example of a problem that involves related rates when t is given and r is not?

One example is a water tank that is being filled at a constant rate of 5 liters per minute. The height of the water in the tank is given as a function of time, h(t) = 3t^2 + 2t + 1. We can use the chain rule to find the rate of change of the height at a specific time t=2 minutes. The formula would be dV/dt = (dh/dt) * (dV/dh), where V represents the volume of water and h represents the height. We can then plug in the values for t and h to solve for the rate of change of the volume.

3) What is the relationship between the given variables in related rates problems?

The given variables are usually related through an equation. For example, in the water tank problem mentioned above, the height of the water is related to the volume through the formula V = Ah, where A is the area of the base of the tank. Understanding this relationship is crucial in setting up the problem and using the chain rule to find the rate of change.

4) How do you know which variable to differentiate with respect to in related rates problems?

The variable you differentiate with respect to depends on the given information and the variable you are trying to find the rate of change for. In most cases, it is the variable that is changing at a constant rate. For example, in the water tank problem, we would differentiate with respect to time since the water is being filled at a constant rate. However, if the problem gave the rate of change of the height, we would differentiate with respect to height.

5) Are there any tips for solving related rates problems when t is given and r is not?

One helpful tip is to draw a diagram or visualize the problem to better understand the relationship between the given variables. Another tip is to carefully label the variables and their units to avoid confusion. It can also be useful to write down the given information and the equation relating the variables before starting to solve the problem. Additionally, practicing with different types of related rates problems can help improve problem-solving skills in this topic.

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