GroupHomework #4

A wire of length 10 inches is to be cut into two pieces. One piece will be used to form a square; the other will be used to form a circle. How should the wire be cut in order to maximize the sum of the areas of the two pieces?

How many tangent lines to the curve NiMvJSJ5RyomJSJ4RyIiIiwmRiZGJ0YnRichIiI= pass through the point (1,5)? At which points do these tangent lines touch the curve? Find equations for all tangents lines and graph them together with the graph of the curve.

NiMqJkkieEc2IiIiIiwmRiRGJkYmRiYhIiI= can be rewritten as NiMsJiIiIkYkKiZGJEYkLCZJInhHNiJGJEYkRiQhIiJGKQ==.

In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equationNiMvLSUlZGlmZkc2JCUiUEclInRHLCYqKCYlInJHNiMiIiEiIiIsJkYvRi8qJkYnRi8mRic2IyUiY0chIiJGNUYvRidGL0YvKiYlJWJldGFHRi9GJ0YvRjU=where NiMmJSJyRzYjIiIh is the birth rate of the fish, NiMmJSJQRzYjJSJjRw== is the maximum population the pond can sustain (called the carrying capacity), and NiMlJWJldGFH is the percentage of the population that is harvested.a) What value of NiMtJSVkaWZmRzYkJSJQRyUidEc= corresponds to a stable population?b) If the pond can sustain 10,000 fish, the birth rate is 5% and the harvesting rate is 4%, find the stable population level.c) If the fish population has reached the stable value and then NiMlJWJldGFH is raised to 5%, what happens to the fish population?

The ACME back-scratcher company can produce NiNJInFHNiI= thousand backstratchers at a cost of NiMvLUkiQ0c2IjYjSSJxR0YmLCoiI2UiIiIqJi1JJkZsb2F0R0kqcHJvdGVjdGVkR0YvNiQiIzosJEYrISIiRitGKEYrRisqJi1GLjYkIiREIiEiJUYrKiRGKCIiI0YrRjMqJi1GLjYkIiImLCRGPkYzRisqJEYoIiIkRitGKw==, where cost is measured in thousands of dollars. a) What is the cost of producing 15 thousand backsratchers?Unfortunately, after producing all those back-scratchers, they have to sell them. In order to sell NiNJInFHNiI= thousand back-scratchers, they have to set the price to NiMvLUkiUEc2IjYjSSJxR0YmLCYiIigiIiIqJi1JJkZsb2F0R0kqcHJvdGVjdGVkR0YvNiQiIiMhIiNGK0YoRishIiI=, where price is measured in dollars. b) Explain why NiMtSSJQRzYiNiNJInFHRiU= decreases as q increases.Revenue is equal to the quantity sold multiplied by the price sold at.c) Find the revenue from selling 15 thousand back-scratchers.Profit is equal to Revenue minus Cost.d) Find the profit from producing and selling 15 thousand back-scratchers.e) Find the production level which will maximize profit for the ACME back-scratcher company. What will the price be at this production level? What will the revenue be?