Find how fast the radius of a circle get smaller as its area gets smaller

In summary, the conversation discusses finding the rate at which the radius of a circle decreases when its area is 75*pi cm^2 and decreasing at a rate of 2*pi cm^2/s. The correct answer is -1/sqrt(75) cm/s, found by using the formula dr/dt = 1/(2*pi*r) dA/dt and evaluating it at the given values.
  • #1
Sennap
6
0

Homework Statement


(Translated, sorry my English) Find how fast does the radius of a circle get smaller when its area is 75*pi cm^2 and it (the area) gets smaller at a rate of 2*pi cm^2/s?

Homework Equations


Not really any given equations, but I used these:
A = r^2 * pi
s = v*t + s0 (If v is a constant)
t=A/Va (where t is the time when A=0)

The Attempt at a Solution


From now on I'll call area, A = 75*pi cm^2 and the rate v_a = 2*pi cm^2/s.
I gave myself that the speed (v) was constant, that shouldn't really change the answer.
So s = dv/dt = v*t + s0

By using A = r^2 * pi we know that:
r^2 = A/pi
r = sqrt(A/pi) and since at t = 0 then A = 75*pi so
r = sqrt(75) at t = 0, let's call r from now on s.

Let's find the time until A = 0, and therefore s=0
t = A/v_a = 75*pi/(2*pi) = 37,5 sec

So in 37,5 sec the circle area will be zero with is radius. Therefore we can see that:
s = v*t + s0
0 = v*37,5 + sqrt(75)
-sqrt(75) = v*37,5
v = -sqrt(75)/37,5
v = -2*sqrt(75)/75
v = -2/sqrt(75)

So my final answer was v = -2/sqrt(75) (or v = -2/(5*sqrt(3))But according to my teacher the correct answer is v = -1/sqrt(75).--
Here's how my teacher did it:

We know that F(t) = pi * r(t)^2 and
(1) F'(t0) = 2*pi*r(t0)*r'(t0) = -2*pi
(2) F(t0) = pi*r(t)^2 = 75*pi
for a specific t0.

According to (2) we know that r(t0) = sqrt(75) = 5*sqrt(3) and according to (1) we get r'(t0) = -1/r(t0) = -1/sqrt(75)

--
He called my solution "a waste of time" although he said it should work too, but couldn't explain to me why I didn't get the same answer.

Where did I go wrong (or even better, where did he go wrong?)
 
Physics news on Phys.org
  • #2
The area of the circle is A(r) = pi r2. Assuming that both A and r are differentiable functions of t, and differentiating with respect to t, you get
dA/dt = 2pi r dr/dt

Solve this equation for dr/dt. That will give you dr/dt for any arbitrary t.
Now evaluate dr/dt at the time when A is 75 pi cm2 and when dA/dt is -2 pi cm2/sec. You will need to find the radius r for which the area A is 75 pi cm2 to do this.
 
  • #3
Thank you for a quick reply!

dr/dt = 1/(2*pi*r) dA/dt and dA/dt = -2*pi so
dr/dt = -1/r

And like I showed before r = sqrt(75) so the final answer is indeed:
dr/dt = -1/sqrt(75)**SOLVED**
Thank you again. I understand your solution a lot better than my teacher's.
 

1. How is the radius of a circle related to its area?

The radius of a circle is directly proportional to its area. This means that as the radius increases, the area also increases and vice versa.

2. What is the formula to find the area of a circle?

The formula for finding the area of a circle is A = πr², where A is the area and r is the radius of the circle. This formula can also be rearranged to find the radius given the area, which is r = √(A/π).

3. How does the radius affect the rate of decrease in the circle's area?

The radius has a direct impact on the rate at which the circle's area decreases. As the radius gets smaller, the area decreases at a faster rate. This is because the area is proportional to the square of the radius, so a small change in the radius results in a larger change in the area.

4. Can you provide an example of how the radius affects the rate of decrease in a circle's area?

For example, if we have a circle with a radius of 10 cm, its area would be 314.16 cm². If we were to decrease the radius to 5 cm, the area would decrease to 78.54 cm². This is a decrease of 235.62 cm², which is much larger than the initial decrease of 5 cm in the radius.

5. Is there a limit to how small the radius and area of a circle can get?

Technically, the radius and area of a circle can approach zero, but they can never actually reach zero. This is because a circle with a radius of zero would not be a circle anymore, it would just be a point. However, for practical purposes, we can consider the radius and area of a circle to be negligible when they are extremely small.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
808
  • Calculus and Beyond Homework Help
Replies
5
Views
222
  • Calculus and Beyond Homework Help
Replies
6
Views
851
  • Calculus and Beyond Homework Help
Replies
16
Views
475
  • Calculus and Beyond Homework Help
Replies
34
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
742
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
901
Back
Top