Please answer each part in the question and show your work, please

In this problem you will derive a partial differential equation for car traffic on a road. We will be assuming that 1. the road only has one lane, so all cars move in the same direction and no passing is permitted; and 2. there are no intersections or ramps onto or off of the road, so cars cannot enter or leave the road. All cars are moving in the positive x direction, from left to right. (a) AC(r, t) is the number of cars from position a to position r + Ar on the road at time t. r(x, t) is the rate of cars passing position x, from left to right, at time t. Explain why the equation AC(x, t + At) = AC(x, t) - r(x + Ax, t)At + r(x, t) At describes how the number of cars in the range x - x + Ar changes with time. (b) c(x, t) is the linear car density on the road, such that AC(x, t) = c(x, t)Ax. By substituting in this expression for C(x, t), and using the definitions of the partial derivatives fx = lim f(x + Ax, t) - c(f, t) Ar+0 Ar ft = lim c(x, t + At) - c(x, t) At-+0 At show C = -72. [Hint: you can take limits of more than one quantity] (c) The rate at which cars pass a position a depends on the number of cars on the road, r(x, t) = B(c(x, t)). The dependence of r on c involves making assumptions. We will simply assume that r is proportional to c, such that r(x, t) = woc(x, t), where we is a constant. Using this proportionality assumption, show that c = -woCr. (d) Is c(x, t) = v(x - wot) is a solution to the partial differential equation of = -Woc? Prove your