Question

The hypotenuse AB of a right triangle ABC is a constant 10 meters, and one leg, AC, is decreasing at the rate of 3 meters per second. Find the rate, in square meters per second, at which the area is changing when AC = 8 meters.

Answer 1

Does anyone know how to solve this?

Answer 2

Let a be the side across from angle A, b be the side across angle B, let c be the side across angle C

the Pythagorean theorem is a^2 + b^2 = c^2

Differentiate both sides wrt t

2a (da/dt) + 2b (db/dt) = 2c (dc/dt)

Since the hypotenuse c is constant, dc/dt is 0. Leg AC (aka side b) is decreasing at 3 m/s so db/dt is -3

Plug in the appropriate quantities and solve for da/dt

Answer 3

From there, area = 1/2(ab)
Differentiating wrt t

dA/dt = (1/2) [a(db/dt) + b(da/dt)]

Since you know that side AC (aka side b) is 8, and the hypotenuse is 10, you can solve for a using the Pythagorean theorem. Finally, plug in da/dt and db/dt from before and evaluate dA/dt

Answer 4

thank you that helped so much