EmojiCon, Inc. is a (fictitious) service company that develops new mass-customized Emojis that look like the customers. They're trying to decide the best price to charge for their service to maximize their expected profits. With the labor, time, and computational resources that go into serving customers' Emoji needs, it costs them a random amount (per customer) that depends on the price charged since customers who pay more expect more (on average). The marginal cost EmojiCon experiences is given by:

c(p)=6exp(p2)+

is a random variable with a normal distribution. It has a mean equal to 0 and a standard deviation that depends on the price charged. The equation for the standard deviation is given by:

(p)=116exp(p2)

The demand (i.e., the number of customers) is also random and depends on the price:

d(p)=4000100p+

is a random variable with a normal distribution. It has a mean equal to 0 and a standard deviation that depends on the price charged as follows:

(p)=80020p

EmojiCon would like to satisfy all of the demand. However, they only have enough workers to serve 1875 customers. Thus, to get the correct number of sales for the purposes of calculating the revenue, you must recall that:

number of units sold=min{demand,capacity}

a. What is the optimal price for EmojiCon to charge each customer for their service (round to the nearest cent)?

b. For the optimal price, what's the expected value for profits (rounded to the nearest whole dollar)?

c. A manager at EmojiCon wants to know more about the riskiness of this optimal price before proceeding. You assure the managerby telling the manager the probability that the profits exceed $25,000. What is the probability the profits exceed $25,000 (round your answer to the nearest hundredth

d. The owner of EmojiCon is thinking about whether to pay for a new software platform that could increase capacity by 25 units (so the new capacity would be 1900). The software platform costs $30. By what dollar amount would profits increase by switching to the new software platform?