Calculus Question

Question

Two sides of a triangle have lengths 7 m and 16 m. The angle between them is increasing at a rate of 0.04radians /min. How fast is the length of the third side increasing when the angle between the sides of fixed length is π/3 radians.

Answer 1

Use the Law of Cosines to establish a relationship between θ (the included angle) and the 3rd side.

Let x represent the length of the side opposite the included angle.

then:

x² = 7² + 16² - 2(7)(16)cosθ

Differentiate implicitly with respect to t (time).

2x dx/dt = -224 (-sinθ) dθ/dt

So,

dx/dt = [224sinθ/2x]dθ/dt

When θ=π/3, use the Law of Cosines to determine the length of side x at that time (left to you).

dx/dt = [112sin(π/3)/(value of 3rd side) ] 0.04

Use the slider on GivenAngle at this graph to see the value of side c increasing as the included angle increases.

desmos.com/calculator/o9dwrcmae7