Redbox and several movie studios have decided to sign a revenue$-$sharing contract for DVDs$.$Each DVD costs the studio $$5$ to produce. The DVD will be sold to Redbox for $$8.$ Redbox, inturn, prices a DVD at $$20.$ At this selling price, demand is assumed to be normally distributed,with a mean of $3,500$ and a standard deviation of $1,200.$ Any unsold DVDs are discounted to $$3,$and all sell at this price. Redbox will share $25\%$ of the revenue with the studio, keeping $75\%$ foritself.

a$)$ How many DVDs should Redbox order?

b$)$ How many DVDs does Redbox expect to sell at a discount?

c$)$ What is the profit that Redbox expects to make?

d$)$ What is the profit that the movie studio expects to make