QUERY 21

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course MTH 271

4/13 4:26pm

021. `query 21*********************************************

Question: `q **** problem 1 7th edition Query 2.8.4 dy/dt for (3,4) with x'=8; dx/dt for (4,3) with y'=-2 ****

What are your solutions?

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Your solution:

equation: x^2 + y^2 = 25

derivative: 2x(dx/dt) = 2y(dy/dt) = 0

replace values with given values:

2(3)(8) + 2(4)(dy/dt) = 0 → 48 + 8(dy/dt) = 0 → 8(dy/dt) = -48 → dy/dt = -48/8 = -6

dy/dt of x^2 + y^2 = 25 at (3,4) is -6.

replace for the second set of values:

2(4)(dx/dt) + 2(3)(-2) = 0 → 8(dx/dt) - 12 = 0 → 8(dx/dt) = 12 → dx/dt = 12/8 = 3/2

dx/dt of x^2 + y^2 = 25 at (4,3) is 3/2.

confidence rating #$&*:

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Given Solution:

`a At (3,4) you are given dx/dt as x ' = 8.

Since 2x dx/dt + 2y dy/dt = 0 we have

2(3) * 8 + 2 * 4 dy/dt = 0 so

dy/dt = -48/8 = -6.

At (4,3) you are given dy/dt as y' = -2. So you get

2 * 4 dx/dt + 2 * 3 * -2 = 0 so

8 dx/dt - 12 = 0 and therefore

8 dx/dt = 12. Solving for dx/dt we get

dx/dt = 12/8 = 3/2. **

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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Question: `q **** problem 2 7th edition Query 2.8.6 roc of volume if r increases at rate 2 in/min, if r= 6 in and if r = 24 in **** What is the rate of volume change if r is increasing at 2 inches / minute?

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Your solution:

The derivative of V = (4/3)*(pi)*(r^3) is:

dV/dt = 4*(pi)*(r^2)*(dr/dt)

dr/dt is given as 2 in/min

When r = 6, dV/dt = 4(pi)(6^2)(2) → dV/dt = 288(pi), or ~904.8

When r = 24, dV/dt = 4(pi)(24^2)(2) → dV/dt = 4608(pi), or ~14,476.5

confidence rating #$&*:

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Given Solution:

`a The shape is a sphere. The volume of a sphere, in terms of its radius, is

V = 4/3 `pi r^3.

Taking the derivative with respect to t, noting that r is the only variable, we obtain

dV/dt = ( 4 `pi r^2) dr/dt

You know that r increases at a rate of 2 in / min, which means that dr/dt = 2.

Plugging in dr/dt = 2 and r = 6 gives 4 pi (6^2) * 2 = 288 pi = 904 approx.

Plugging in dr/dt = 2 and r = 24 gives 4 pi (24^2) * 2 = 4 pi (576)(2) = 4608 pi = 14,476 approx.. **

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Self-critique (if necessary): OK

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Self-critique Rating: 3

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