If x2 + y2 = 100 and dy/dt = 3, find dx/dt when y = 8.
your answer is wrong :(
dx/dt = -4 and 4
To find dx/dt when y = 8, we need to establish a relationship between x and y.
We are given the equation x^2 + y^2 = 100. We can differentiate both sides of this equation with respect to time t using the chain rule:
d/dt (x^2 + y^2) = d/dt (100)
Differentiating x^2 with respect to t gives us 2x(dx/dt), while differentiating y^2 with respect to t gives us 2y(dy/dt). The derivative of a constant (100 in this case) is 0. Therefore, the equation becomes:
2x(dx/dt) + 2y(dy/dt) = 0
Substituting dy/dt = 3 and y = 8 into the equation, we get:
2x(dx/dt) + 2(8)(3) = 0
Simplifying further:
2x(dx/dt) + 48 = 0
Now we can solve for dx/dt:
2x(dx/dt) = -48
dx/dt = -48 / 2x
To find dx/dt when y = 8, we need to find the corresponding value of x. Substituting y = 8 into x^2 + y^2 = 100:
x^2 + 8^2 = 100
x^2 + 64 = 100
x^2 = 100 - 64
x^2 = 36
Taking the square root of both sides:
x = ±6
Since we are given dy/dt = 3, we assume that x > 0 (since we are interested in the rate of change when y = 8). Therefore, x = 6.
Substituting x = 6 into the equation dx/dt = -48 / 2x:
dx/dt = -48 / 2(6)
dx/dt = -48 / 12
dx/dt = -4
Therefore, when y = 8, dx/dt = -4.
I assume you meant to write
x^2 + y^2 = 100.
That is the equation of a circle centered at the origin with a radius of 10.
dx/dt = (dy/dt)/(dy/dx) can be used to calculate dx/dt.
When y = 8, x^2 = 36 and x = +6 or -6
I'll just use the +6 answer for x, but there are two possible answers here.
2x + 2y*(dy/dx) = 0
dy/dx = -x/y
Since dy/dt = 3,
dx/dt = 3/(-6/8)= -4