Optimization? A right triangle has legs of len ? gth ?h? a? and a hypotenuse of length 4 (see figure). It is revolved about the leg of le? ng? to sweep out a right circular cone. What val?ues of? ? and ? maximize the volume of the cone?
Solution Step 1 In this problem we have given the sides of a right triangle h and r and hypotenuse of length 4. Now if the triangle is revolved about the side h to form a right circular cone as in the figure, we have to find for what values of h and r the volume of the cone is maximized. In order to find the values of h and r so that the the volume of the cone increases it is enough to find the critical points regarding h and r . Critical point: An interior point cof the domain of a function f at which f (c) = or f c)fails to exist is called a critical point of f Step 2 In order to solve this problem we will be using the following formulas Volume of the cone V = 1 r h 3 In a right triangle the square of the hypotenuse is equal to the sum of the square of the other two sides. 2 2 In this case we get 4 = r + h Consider 4 = r + h2 2 2 2 2 r = 4 h 2 Therefore substituting the value of r in volume of the cone formula, we get 2 2 The volume of the cone, V = (4 3 )h V = (16 h )h 2 3 1 3 V = 3(16h h )