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a circle is inscribed in a square. the circumference of the circle is increading at a constant rate of 6 inches per second. As the circle expands, the aquare expands to maintain the condition of tangency. find th rate at which the perimeter of the square is increasing

Respuesta :

Answer:

The rate at which Perimeter of the square is increasing is [tex]\frac{24}{\pi} \ in/secs[/tex].

Step-by-step explanation:

Given:

Circumference of the circle = [tex]2\pi r[/tex]

Rate of change of in circumference = 6 in/secs

We need to find the rate at which the perimeter of the square is increasing

Solution:

Now we know that;

[tex]\frac{d(2\pi r)}{dt} =6\\\\2\pi\frac{dr}{dt}=6\\\\\frac{dr}{dt}=\frac{6}{2\pi}\\\\\frac{dr}{dt}=\frac{3}{\pi}[/tex]

Now we know that;

side of the square= diameter of the circle

side of the square = [tex]2r[/tex]

Now Perimeter of the square is given by 4 times length of the side.

[tex]P=4\times 2r =8r[/tex]

Now we need to find the rate at which Perimeter is increasing so we will find the derivative of perimeter.

[tex]\frac{dP}{dt}= \frac{d(8r)}{dt}\\\\\frac{dP}{dt}= 8\times\frac{dr}{dt}[/tex]

But [tex]\frac{dr}{dt} =\frac{3}{\pi}[/tex]

So we get;

[tex]\frac{dP}{dt}= 8\times\frac{3}{\pi}\\\\\frac{dP}{dt}= \frac{24}{\pi}\ in/sec[/tex]

Hence The rate at which Perimeter of the square is increasing is [tex]\frac{24}{\pi} \ in/secs[/tex].

facundo3141592

We want to find the rate at which the perimeter of the square increases. We will see that the perimeter increases at a rate of (24/3.14) in/min

Let's see how to solve the problem:

We have a circle inscribed in a square, this means that the diameter of the circle is equal to the side length of the square.

Now, the first thing we must do is find the rate at which the diameter of the circle increases.

We know that the circumference increases at a rate of 6 in/min.

And the circumference of a circle of diameter D is given by:

C = 3.14*D

Then if the circumference increases at a rate of 6 in/min, then the diameter increases at a rate of (6/3.14) in/min.

This means that the side length of the square also increases at this rate, and the perimeter of the square is four times the side length, then the perimeter of the square increases at four times the above rate:

4*(6/3.14) in/min = (24/3.14) in/min

So the perimeter increases at a rate of (24/3.14) in/min

If you want to learn more about geometry, you can read:

https://brainly.com/question/14283575

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