The length of each leg of an isosceles right triangle is increase at the rate of 2 mm/s. How fast is the hypotenuse increasing at the moment when each leg is 10 mm?
h^2 = 2 x^2
2 h dh/dx = 4 x
dh/dx = 2 x/h
dh/dt = dh/dx * dx/dt = 2 (x/h) dx/dt
but x/h = 1/sqrt 2
so
dh/dt = sqrt 2 * dx/dt
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but more simply
h = x sqrt 2
dh/dx = sqrt 2
To find the rate at which the hypotenuse is increasing, we can use the Pythagorean theorem. In an isosceles right triangle, the two legs are equal in length, so let's call the length of each leg "x" and the length of the hypotenuse "h".
According to the Pythagorean theorem, we have:
x^2 + x^2 = h^2
Simplifying this equation, we get:
2x^2 = h^2
Now, to find the rate at which the hypotenuse is increasing (dh/dt) when each leg is 10 mm, we need to differentiate the equation with respect to time (t):
d(2x^2)/dt = d(h^2)/dt
2 * d(x^2)/dt = 2h * dh/dt
Since we are given that dx/dt (rate at which the legs are increasing) is 2 mm/s, we have:
2 * d(x^2)/dt = 2h * dh/dt
Differentiating x^2 with respect to t, we get:
2x * dx/dt = 2h * dh/dt
Plugging in the given values of dx/dt = 2 mm/s, and x = 10 mm (since each leg is 10 mm), we obtain:
2 * (10 mm) * (2 mm/s) = 2h * dh/dt
Simplifying further:
40 mm * mm/s = 2h * dh/dt
Now, since h is not given, we need to find its value. Since the triangle is an isosceles right triangle and each leg is 10 mm, we can use the Pythagorean theorem to find the value of h:
x^2 + x^2 = h^2
(10 mm)^2 + (10 mm)^2 = h^2
100 mm^2 + 100 mm^2 = h^2
200 mm^2 = h^2
Taking the square root of both sides:
h = sqrt(200) mm
Now, let's plug in the value of h into our equation:
40 mm * mm/s = 2 * (sqrt(200) mm) * dh/dt
Simplifying again:
40 = 2 * sqrt(200) * dh/dt
To find the value of dh/dt (the rate at which the hypotenuse is increasing), we need to solve for it:
dh/dt = 40 / (2 * sqrt(200))
Calculating this expression, we get:
dh/dt ≈ 5.6569 mm/s
Therefore, the hypotenuse is increasing at a rate of approximately 5.6569 mm/s when each leg is 10 mm.