I need solution for problem 1A, 1D, and 1E for me to understand. Thank you!!

Problem 1: Market Equilibrium and Total Surplus [3.5 points] Consider the market for coal. Let the market demand function for coal be Q = 1000 - 500p, where p is the price of coal. Suppose the U.S. is covered with many small local coal mines, all of which have the same production process and are producing identical products. Thus, the U.S. coal industry is a perfectly competitive market, with each mine acting as if they can sell as much (or as little) as they want without affecting the price (i.e., they are price takers). Consider a single coal mine whose cost of mining q units of coal is represented by C(q) = q2. Thus, a single mine earns profits T (q) = p x q - q. 1.A. [0.5 points] Assume that a single coal mine chooses coal production q to maximize profits. Find the amount of coal q" that maximizes a caol mine's profits. 1.B. [0.5 points] Suppose that the government limits the number of mines in the U.S to 1,000 via a permit program. What is the industry supply curve? And what are the market equilibrium quantity produced Q* and price of coal p*? 1.C. [0.5 point] What are the producer and consumer surplus in the U.S. market for coal? 1.D. [1 point] It tuns out that burning coal has some negative social costs-it produces greenhouse gasses that contribute to global warm- ing. Suppose that the total social damages are equal to D = 6Q, where o = 1. Determine the socially optimal level of coal Q** that should be produced. 1.E. [1 point] Suppose that the government wants to restrict the amount of coal produced to achieve the socially optimal level that you found in Part E. The government has decided to do this by reducing the number of coal mining permits. How many permits should it issue