Philip P. answered • 01/01/19
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For an isosceles right triangle with sides s, the hypotenuse is s√2. The rate of change of the hypotenuse is d(s√2)/dt = √2·ds/dt.
A = (1/2)s2
dA/dt = d(s2/2)/ds · ds/dt
dA/dt = s · ds/dt
Now dA/dt = 12 cm2/sec when s = √32 = 4√2
12 = 4√2 · ds/dt
3 cm/sec = √2 · ds/dt = the rate of change of the hypotenuse
Aneesh S.
When Stacey M. said take the antiderivative of each side, they meant to type "take the derivative", not antiderivative. There is no integration in this problem, derivative of 2s^2 is 4s.
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04/11/24
Stacey M.
For anyone looking up this question on a later date. Philip did not finish the question. ds/dt is 3/sqrt2, but that is the not the rate the length the hypotenuse is changing. You know s = 4sqrt2 , dA/dt = 12 cm/sec , and now you also know ds/dt = 3/sqrt2 , but it was asking for the hypotenuse. The area formula has nothing to do with the hypotenuse, but since you have a right triangle you can use the pythagorean theorem. If you substitute sqrt32 into pythagorean theorem you get that the hypotenuse is 8 when the length of the legs of sqrt32, but that is still not what it is asking for. It is asking for the change in the hypotenuse. So if you use the pythagorean as 2s^2 = c^2 and take the antiderivative of each side, you get 4s ds/dt = 2c dc/dt with dc/dt representing the change in the hypotenuse in terms of time. Substitute in 4(4sqrt2)(3/sqrt2) = 2(8) dc/dt Solve for dc/dt and you get 3.05/10/19