Question

Two sides of a triangle have lengths 4 m and 5 m. The measure of the angle between them is increasing at a rate of 0.05 rad/sec. Find the rate at which the area of the triangle increases, when the angle between the sides (fixed) has measure π/3.

Answer 1

solution : from properties of triangle,

area of triangle , $A=\frac{1}{2}absin\theta$

here, a and b are side length of triangle and θ is angle between a and b.

here, a = 4, Ans b = 5 and $\frac{d\theta}{dt}$=0.05rad/s

now, A = 1/2 × 4 × 5 $sin\theta$

$A=10sin\theta$

differentiate with respect to time,

dA/dt = $10cos\theta\times\frac{d\theta}{dt}$

= 10 × cosπ/3 × 0.05

= 10 × 1/2 × 0.05 m²/s

= 0.25 m²/s