Respuesta :
Answer:
The area of the triangle is increasing at a rate 1.2 m²/s.
Step-by-step explanation:
Given sides of a triangle are 8 m and 10 m in length.
The angle between the sides of the triangle is increasing at a rate 0.06 rad/s.
It means [tex]\frac{d\theta}{dt}=0.06[/tex]
If the sides of a triangle are a and b and θ is the angle of the sides.
The area of the triangle is A = [tex]\frac{1}{2}ab sin\theta[/tex]
Here, a = 8 m and b = 10 m
[tex]A= \frac{1}{2}\times 8 \times 10 sin \theta[/tex]
[tex]\Rightarrow A = 40 sin \theta[/tex]
Differentiating with respect to t
[tex]\frac{dA}{dt}= 40 cos\theta \frac{d\theta}{dt}[/tex]
putting the value of [tex]\frac{d\theta}{dt}[/tex]
[tex]\Rightarrow \frac{dA}{dt}|_{\theta=\frac{\theta}{3}}= 40 cos\frac{\pi}{3} \times 0.06[/tex]
[tex]= 40 \times \frac{1}{2} \times 0.06[/tex]
=1.2 m²/s
The area of the triangle is increasing at a rate 1.2 m²/s.
