4. Let x be the length of one of the equal sides of an isosceles triangle, and let q be the angle between them. If x is increasing at the rate 1/12 m/hr, and q is increasing at the rate of p /180 radians/hr, then at what rate, in m2/hr, is the area of the triangle increasing when x = 12 m and q = p /4?

 

A. 36(21/2)

B. (73/2)(21/2)

C. (31/2/2)+(p /5)

D. 21/2((1/2)+(1/5)p )

E. 21/2(6+(1/5)p )

 

 

Solution: The answer is D

 Consider the following isosceles triangle;

  

 

Where h is the height of the triangle, b is the base, and q is the angle between the equal lengths, X. Since the given triangle is isosceles we can define b and h as follows,

So that the area, A, is defined as,

 

Differentiating both sides of the above equation with respect to time t yields,

Since, dx/dt = 1/12 and dq /dt = p /180. Then by substitution,

Finally, when x = 12 and q = p /4,