Question

If x2 + y2 = 100 and dy/dt = 9, find dx/dt when y = 8. (Enter your answers as a comma-separated list.)

Answer 1

-12, 12

Explanation:

Given x² +y² = 100, dy/dt = 9, and y = 8, you want to find dx/dt.

Derivative

Differentiating the given equation with respect to t, we have ...

2x dx/dt +2y dy/dt = 0

Solving for dx/dt, we find ...

dx/dt = -y/x·dy/dt

For the given values of y and dy/dt, this is ...

dx/dt = -8/x·9 = -72/x

X-value

The value of x when y = 8 can be found from the original equation:

x² +8² = 100

x² = 36 . . . . . . . . subtract 64

x = ±6 . . . . . . . . square root

Then the desired values of dx/dt are ...

dx/dt = -72/6 = -12

dx/dt = -72/(-6) = 12

The values of dx/dt are -12, 12.

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Answer 2

dx/dt = -12, 12

Explanation:

The question asks us to find 'dx/dt' when y = 8, given that x² + y² = 100 and dy/dt = 9. The approach involves differentiating both sides of the equation x² + y² = 100 with respect to time 't' and then solving for 'dx/dt' using the given values.

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Differentiating Both Sides with Respect to Time
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We start with the equation:

`\Longrightarrow x^2 + y^2 = 100`

Taking the derivative of both sides with respect to time 't' gives:

`\Longrightarrow \displaystyle (d)/(dt)[x^2] + (d)/(dt)[y^2] = (d)/(dt)[100]`

Using the chain rule, this simplifies to:

`\Longrightarrow \displaystyle 2x(dx)/(dt) + 2y(dy)/(dt)= 0`

Further simplification leads to:

`\Longrightarrow \displaystyle x(dx)/(dt) + y(dy)/(dt)= 0`

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Substituting Given Values and Solving for 'dx/dt'
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We are given that dy/dt = 9 and y = 8. To find 'x' when y = 8, we use the equation 'x² + y² = 100':

`\Longrightarrow x^2 + (8)^2 = 100\\\\\\\\\Longrightarrow x^2 + 64 = 100\\\\\\\\\Longrightarrow x^2 = 36\\\\\\\\\thereforex = \pm 6`

Assuming x = 6 (the logic applies if x = -6 as well), we can now substitute these values into x dx/dt + y dy/dt = 0:

For x = 6:

`\Longrightarrow \displaystyle (6)(dx)/(dt) + (8)(9)= 0\\\\\\\\\Longrightarrow \displaystyle 6(dx)/(dt) + 72= 0\\\\\\\\\Longrightarrow \displaystyle 6(dx)/(dt) = -72\\\\\\\\\therefore \boxed{(dx)/(dt)=-12}`

For x = -6:

`\Longrightarrow \displaystyle (-6)(dx)/(dt) + (8)(9)= 0\\\\\\\\\Longrightarrow \displaystyle -6(dx)/(dt) + 72= 0\\\\\\\\\Longrightarrow \displaystyle -6(dx)/(dt) = -72\\\\\\\\\therefore \boxed{(dx)/(dt)=12}`

Thus, dx/dt = -12, 12.