Suppose that x=x(t) and y=y(t) are both functions of t. If x^2+y^2=26 and dx/dt=−2 when x=1 and y=5, what is dy/dt?
help??????
Ah, the good ol' world of calculus. Don't worry, I'll put on my mathematical clown nose and help you out!
First, let's work with the equation x^2 + y^2 = 26. We know that dx/dt = -2, and we want to find dy/dt.
To do that, we need to differentiate both sides of the equation with respect to t using the chain rule. So, we get:
2x * dx/dt + 2y * dy/dt = 0
Now, we can substitute the given values for x, y, and dx/dt:
2(1)(-2) + 2(5)(dy/dt) = 0
Simplifying this equation, we have:
-4 + 10(dy/dt) = 0
Now, let's isolate dy/dt:
10(dy/dt) = 4
dy/dt = 4/10
So, dy/dt = 2/5.
Tada! We've arrived at the answer. Dy/dt is equal to a proud 2/5. Hope I was able to add a little mathematical amusement to your day!
To find dy/dt, we need to use the given information and differentiate the equation x^2 + y^2 = 26 with respect to t. Following the chain rule, we can write:
2x * dx/dt + 2y * dy/dt = 0
Since dx/dt is given as -2 and we need to find dy/dt, we can substitute the known values into the equation:
2(1)(-2) + 2(5) * dy/dt = 0
Simplifying the equation:
-4 + 10 * dy/dt = 0
Rearranging the terms:
10 * dy/dt = 4
Finally, solving for dy/dt:
dy/dt = 4 / 10
dy/dt = 0.4
Therefore, dy/dt is equal to 0.4.
To find dy/dt, we need to differentiate the equation x^2 + y^2 = 26 with respect to t.
Differentiating both sides of the equation, we get:
2x(dx/dt) + 2y(dy/dt) = 0
Now, we substitute the given values dx/dt = -2, x = 1, and y = 5 into the equation:
2(1)(-2) + 2(5)(dy/dt) = 0
Simplifying this equation, we have:
-4 + 10(dy/dt) = 0
Now, let's solve for dy/dt:
10(dy/dt) = 4
dy/dt = 4/10
dy/dt = 2/5
Thus, the value of dy/dt is 2/5 or 0.4.
differentiate implicitly
2x dx/dt + 2y dy/dt = 0
sub in the values
2(1)(-2) + 2(5)dy/dt = 0
dy/dt = 4/10 - 2/5