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Section 5

Applications II - Business and Economic Optimization Problems

A producer finds that demand for his commodity obeys a linear demand equation $p+2 x=100$ where $p$ is in dollars and $x$ in thousands of units.(a) Find the level of production that will maximize revenue. If the producer's costs are given by $C(x)=2+3 x,$ what should his level of production be to maximize profits?

Find the minimum average cost if the total cost function is $C(x)=x^{2}+5 x+9$

Minimize the average cost if the total cost function is $C(x)=3 x^{2}+200 x+12$

Find the maximum revenue if the demand equation is: (a) $p=100 /(x+1)$(b) $p=\frac{\left(50-x^{4}\right)}{(x+2)} ;$ (c) $p+x=75 ;$ (d) $p^{2}+x^{2}=100$(e) $p^{2}-x=100 ;(\mathrm{f}) p^{2}+x^{2}=50$

If the cost equation in Exercise I is $C(x)=0.5 x^{2}+x+1,$ what price should be charged to maximize profit?

A fur dealer finds that when coats sell for $\$ 4000,$ monthly sales are 6 coats. When the price increases to $\$ 5000,$ the demand is for 5 coats. Assume that the demand equation is linear. (a) Find the demand and revenue equations.(b) If overhead is $\$ 2500$ per month and the production cost per coat is $\$ 2000$, find the cost equation and profit equation. (c) Find the level of production that maximizes profit.

Given the demand equation $p=100 /(x+1),$ and a cost equation $C(x)=x+1,$ find the maximum profit.

If the demand equation is $p=\sqrt{9}-x+3,$ and the cost equation is $C(x)=3 x+1,$ find the maximum profit.

A diamond dealer finds that the demand for flawless diamonds is governed by $p=-x^{2}+4 x,$ where $x$ is in carats and $p$ is in thousands of dollars.(a) For what values of $x$ is this a possible demand equation? (b) Find the number of carats to be made available to maximize revenue. What is the price and revenue for this amount?

Each month an automobile dealer makes a profit of $\$ 200$ on each car that she sells if not more than 50 cars are sold. For every car above 50 that she sells her profit per car is decreased by $\$ 2 .$ How many cars should the dealer sell monthly to maximize her profit?

An orchard contains 300 peach trees with each tree yielding 800 peaches. For each five additional trees planted, the yield per tree decreases by 10 peaches. How many trees should be planted to maximize the total yield of the orchard?

Find the elasticity of demand for each of the following demand equations.(a) $x+4 p=200 ;$ (b) $x=\sqrt{100-p} ;$ (c) $x=100(p+1)^{2}$

Find the elasticity of demand at the indicated price for each of the following.(a) $p+4 x=80, p=40 $(b) $10 p+x=500, p=46$(c) $4 p+x=100, p=0 $ (d) $p x=8, p=4$(e) $p x^{2}=12, p=3 $(f) $p=\sqrt{9-x}, p=2$

Suppose in Exercise 5 that the producer is subjected to a tax of $\$ 10$ per thousand units. What should his production level be in order to maximize profits?

In Exercise 6 if the government applies a luxury tax of $20 \%$ per coat, how are profits affected?

In Exercise $5,$ what tax should the government impose on the producer if it wishes to maximize its tax revenue?

In Exercise $6,$ what tax should the government impose on the producer if it wishes to maximize its tax revenue?

Figure 2 indicates the graph of a demand equation, and its tangent line at the point $B$. Show that $\epsilon_{D}=-d_{A B} / d_{B C},$ where $d_{A B}$ is the distance from the point $B$ to the $x$ -intercept $A$ and $d_{B C}$ is the distance from $B$ to the $p$ -intercept $C$

Use the results of the preceding exercise and a scale drawing to find the elasticity of demand for each of the following. Compare the result with the exact answer found by using either equation (1) or (4) . (a) $p+4 x=80$ at $p=40 .(\mathrm{b}) p=\sqrt{9-x}$ at $p=2 .(\mathrm{c}) p / x=10$ at $p=5$

For a linear demand equation show that $\epsilon_{D}=-A p / x .$ Interpret $A$

If a demand equation has the form $x p^{c}=A$ show that $\epsilon_{D}=-c,$ where $c$ and $A$ are constants.

Given a linear demand equation $x=m p+b$. Let $p$ change from $p_{0}$ to $p_{0}+h .$ Show that the ratio of the relative change in $x$ to the relative change in $p$ is exactly equal to the elasticity of demand at $p_{0}$

Suppose $x=D(p)$ is a nonlinear demand equation. Let price change from $p$ to $p+h .$ Show that clasticity of demand is the limit as $h$ approaches zero of the relative change in demand to the relative change in price.

Derive equation (4).