Let x be the length of each of the equal sides of an isosceles triangle and be the angle between these sides. If x is increasing at the rate 1 12 ~m / hour and is increasing at the rate 180 rad / hour, then the rate at which area of the triangle is increasing when x=12 ~m and = 4 is

Step-by-step solution

Area of the isosceles triangle: A = 1 2 x^2 Differentiating with respect to time t using the product rule: dA dt = 1 2 [ 2x dx dt + x^2 d dt ] Substituting x = 12 m , = 4 , dx dt = 1 12 m/hour , d dt = 180 rad/hour : dA dt = 1 2 [ 2 12 1 12 1 2 + 144 1 2 180 ] Simplifying the expression: dA dt = 1 2 [ 2 2 + 144 180 2 ] = 1 2 [ 2 + 4 5 2 ] Combining terms over a common denominator: dA dt = 1 2 10 + 4 5 2 = 5 + 2 5 2 Final simplification yields: dA dt = 2 ( 1 2 + 5 ) m ^2/ hour This matches option B .