(Solved): (22 pts) A lumber yard is located next to a piano school. Sound from the lumber yard impacts the pi ...

(22 pts) A lumber yard is located next to a piano school. Sound from the lumber yard impacts the piano school. The lumber yard's profit function is ${\pi }_L(t)=40t-\frac{1}{2}{t}^2$ where $t$ is the number of trees processed by the lumber yard. The piano school's profit function is ${\pi }_p(h,t)=60h-h^2-\frac{1}{2}ht$ where $h$ is the number of hours the piano school teaches. a. How many trees will the lumber yard process? Show your work. b. How many hours will the piano school teach? Show your work. c. At the solutions to parts (a) and (b), use the partial derivative of each firm's profit with respect to the number of trees processed to demonstrate that the situation described in parts (a) and (b) is inefficient. To achieve efficiency, should the number of trees processed increase or decrease? Clearly demonstrate your logic.