If x^2+y^2=25, find dy/dt when x=3 and dx/dt= -8.
when x = 3
9+y^2 = 25
y^2 = 16
y = ±4
2x dx/dt + 2y dy/dt = 0
dy/dt = -x dx/dt/y
= -3(-8)/±4
= ± 6
You mean
on this circle find vertical component of velocity.
2 x dx/dx + 2 y dy/dx = 0
y dy/dx = -x
y dy/dx * dx/dt = -x * dx/dt
dy/dt = -(x/y) dx/dt
here x = 3
then y = +4 or -4
if y is +, dy/dt is + for -dx/dt
if y is -, dy/dt is - for -dx/dt
so
dy/dt = -(3/4)(-8) = 6 in first quadrant
dy/dt = -(3/-4)(-8) = -6 in quadrant 4
Well, if x^2 + y^2 = 25, then we can differentiate both sides with respect to t using the chain rule. But before we go down that rabbit hole, let's take a moment to appreciate the beauty of this equation: a little Pythagorean love triangle with its hypotenuse equal to 5. Ah, mathematics, bringing harmony to our lives!
Now, where were we? Oh right, differentiating. So, we have 2x(dx/dt) + 2y(dy/dt) = 0. Plugging in the given values, we get: 2(3)(-8) + 2y(dy/dt) = 0.
Simplifying that, we have -48 + 2y(dy/dt) = 0. Solving for dy/dt, we find: dy/dt = 48/2y.
But wait, there's more! We also know that x^2 + y^2 = 25. So, when x = 3, we can substitute that in and get: 9 + y^2 = 25. Solving for y, we get y = ±√16. Taking the positive square root, we have y = 4.
Putting it all together, dy/dt = 48/2(4) = 6. So, when x = 3 and dx/dt = -8, dy/dt = 6. With these numbers, you'll be able to track the rate of change of y as x and t change. Good luck!
To find dy/dt, we need to differentiate the equation x^2+y^2=25 implicitly with respect to time (t), since x and y are considered as functions of t.
Let's differentiate both sides of the equation with respect to t, using the chain rule:
d/dt(x^2) + d/dt(y^2) = d/dt(25)
To find dy/dt, we are given dx/dt, which is -8. So, we can substitute -8 for dx/dt in the equation above:
2x(dx/dt) + 2y(dy/dt) = 0
Substitute x=3 and dx/dt=-8 into the equation:
2(3)(-8) + 2y(dy/dt) = 0
-48 + 2y(dy/dt) = 0
Now, we need to solve for dy/dt. Rearranging the equation:
2y(dy/dt) = 48
Divide both sides by 2y:
(dy/dt) = 48 / (2y)
Now, to find the value of dy/dt when x=3, we need to determine the value of y.
Given the equation x^2 + y^2 = 25, and substituting x=3:
3^2 + y^2 = 25
9 + y^2 = 25
y^2 = 25 - 9
y^2 = 16
Taking the square root of both sides:
y = ±4
Since we are calculating dy/dt, we don't need to consider the positive or negative square root. Let's use y = 4:
(dy/dt) = 48 / (2 * 4)
(dy/dt) = 48 / 8
(dy/dt) = 6
Therefore, dy/dt = 6 when x=3 and dx/dt = -8.