Question

The hypotenuse of a right triangle is 25 ft. If one of the acute angles increases at rate of 4 degrees per second, how fast is the area increasing when the angle is 30 degrees.

Answer 1

let the base of the triangle be x
let its height be y
let the angle at the base be Ø

then
cosØ =x/25 ---> x = 25cosØ
sinØ = y/25 ---> y = 25sinØ

Area = (1/2)xy
= (1/2)(25cosØ)(25sinØ)
= (625/2)sinØcosØ
= (625/4) sin 2Ø , (using sin 2A = 2sinAcosA)

Trig derivatives are only valid if the angle is in radians, so
4° / sec = π/45 rad/sec
30° = π/6

d(area)/dπ= (625/4) cos (2Ø) (2) dØ/dt
= (625/4)(cos π/3) (2) (π/45)
= (625/4)(1/2)(2)(π/45)
= 10.91 ft^2/sec