A company, founded at the end of December, 2019, sells unicorns: the price and quantity of unicorns sold fluctuate over time depending on market pressures like demand. During April 2020, the price of a unicorn was $1000 per unicorn and the quantity of unicorns sold was 5. At the end of April, demand, currently at 5 people, was rising, with respect to time, at a rate of 2 people per month, and price was rising, with respect to demand, at a rate of $10 per person. However quantity sold, with respect to time, was decreasing at a rate of 1 unicorn per month. Let t denote the time, in months, since the end of December 2019,

Let R(t) denote the months revenue, in dollars, at time t,

Let d(t) denote the demand for unicorns, in number of people, at time t,

Let P(d) denote the price, in dollars per unicorn, at demand d, and

Let q(t) denote the quantity sold that month, in number of unicorns, at time t.

(a) Express the given information about the company in terms of the functions d, P, q and their derivatives. Your answer should have 6 components: each of d(t), P(d), q(t) evaluated at a particular value, and then each of d 0 (t), P0 (d), and q 0 (t) evaluated at a particular value.

(b) Find the rate of change of price with respect to time at the end of April, including units, using your answers to part (a). (Recall price is a function of demand, which is in turn a function of time.)

(c) Monthly revenue is computed by multiplying that months price times the quantity sold that month. Find the rate of change of revenue with respect to time at the end of April, including units, using your answers to part (a).