Suppose that it costs $200 per day to search for chanterelle mushrooms at Pt. Reyes National Seashore. On an average day, the total weight of mushrooms M found at Pt. Reyes is M = 100x-x^2 pounds ,where x is the number of people mushroom hunting on that day. Chanterelles can be sold for $60 per pound. How many more people will go mushroom hunting than is socially optimal?

For an overall profit, we need at least 97 people to go mushroom hunting.

Any number of people that is more than the socially optimal number should go mushroom hunting on any given day.

Step-by-step explanation:

The socially optimal number of people that will go mushroom hunting is the number where amount spent to go mushroom hunting equally balances the amount obtained by selling the mushrooms obtained.

If x people go mushroom hunting in a day, the total cost of hunting for that day = 200x

The amount of mushroom obtained is given as

M = (100x - x²) in pounds

The selling price of 1 pound = $60

The cost of M pounds = 60M = 60(100x - x²)

= (6000x - 60x²)

At socially optimal number,

200x = 6000x - 60x²

60x² - 6000x + 200x = 0

60x² - 5800x = 0

x(60x - 5800)

x = 0 or (60x - 5800) = 0

x = 0 or x = (5800/60) = 96.67

Socially optimal number of people = 0 or 96.67

For realistic purposes, we take the socially optimal number of people that went mushroom hunting as 96.67

Any number above this number will result in an overall profit, and any number below it results in an overall loss.

So, for an overall profit, we need at least 97 people to go mushroom hunting.

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