Stewart 8th Section 8.4: Applications to Economics and Biology

The marginal cost function \(C'(x)\) was defined to be the derivative of the cost function. (See Sections 3.7 and 4.7.) The marginal cost of producing \(x\) gallons of orange juice is \(C'(x) = 0.82 - 0.00003 x + 0.000000003 x^2\) (measured in dollars per gallon). The fixed start-up cost is \(C(0) = \$18{,}000\). Use the Net Change Theorem to find the cost of producing the first \(4000\) gallons of juice.

A company estimates that the marginal revenue (in dollars per unit) realized by selling \(x\) units of a product is \(48 - 0.0012 x\). Assuming the estimate is accurate, find the increase in revenue if sales increase from \(5000\) units to \(10{,}000\) units.

A mining company estimates that the marginal cost of extracting \(x\) tons of copper ore from a mine is \(0.6 + 0.008 x\), measured in thousands of dollars per ton. Start-up costs are \(\$100{,}000\). What is the cost of extracting the first \(50\) tons of copper? What about the next \(50\) tons?

The demand function for a particular vacation package is \(p(x) = 2000 - 46 \sqrt{x}\). Find the consumer surplus when the sales level for the packages is \(400\). Illustrate by drawing the demand curve and identifying the consumer surplus as an area.

A demand curve is given by \(p = 450 / (x + 8)\). Find the consumer surplus when the selling price is \(\$10\).

The supply function \(p_S(x)\) for a commodity gives the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will produce more units, so \(p_S\) is an increasing function of \(x\). Let \(X\) be the amount of the commodity currently produced and let \(P = p_S(X)\) be the current price. Some producers would be willing to make and sell the commodity for a lower selling price and are therefore receiving more than their minimal price. The excess is called the producer surplus. An argument similar to that for consumer surplus shows that the surplus is given by the integral \(\displaystyle\int_{0}^{X} [P - p_S(x)] d x\). Calculate the producer surplus for the supply function \(p_S(x) = 3 + 0.01 x^2\) at the sales level \(X = 10\). Illustrate by drawing the supply curve and identifying the producer surplus as an area.

If a supply curve is modeled by the equation \(p = 125 + 0.002 x^2\), find the producer surplus when the selling price is \(\$625\).

In a purely competitive market, the price of a good is naturally driven to the value where the quantity demanded by consumers matches the quantity made by producers, and the market is said to be in equilibrium. These values are the coordinates of the point of intersection of the supply and demand curves.

The sum of consumer surplus and producer surplus is called the total surplus; it is one measure economists use as an indicator of the economic health of a society. Total surplus is maximized when the market for a good is in equilibrium.

A camera company estimates that the demand function for its new digital camera is \(p(x) = 312 e^{-0.14 x}\) and the supply function is estimated to be \(p_S(x) = 26 e^{0.2 x}\), where \(x\) is measured in thousands. Compute the maximum total surplus.

A company modeled the demand curve for its product (in dollars) by the equation \(p = \dfrac{800{,}000 e^{-\dfrac{x}{5000}}}{x + 20{,}000}\). Use a graph to estimate the sales level when the selling price is \(\$16\). Then find (approximately) the consumer surplus for this sales level.

A movie theater has been charging \(\$10.00\) per person and selling about \(500\) tickets on a typical weeknight. After surveying their customers, the theater management estimates that for every \(50\) cents that they lower the price, the number of moviegoers will increase by \(50\) per night. Find the demand function and calculate the consumer surplus when the tickets are priced at \(\$8.00\).

If the amount of capital that a company has at time \(t\) is \(f(t)\), then the derivative, \(f'(t)\), is called the net investment flow. Suppose that the net investment flow is \(\sqrt{t}\) million dollars per year (where \(t\) is measured in years). Find the increase in capital (the capital formation) from the fourth year to the eighth year.

If revenue flows into a company at a rate of \(f(t) = 9000 \sqrt{1 + 2 t}\), where \(t\) is measured in years and \(f(t)\) is measured in dollars per year, find the total revenue obtained in the first four years.

If income is continuously collected at a rate of \(f(t)\) dollars per year and will be invested at a constant interest rate \(r\) (compounded continuously) for a period of \(T\) years, then the future value of the income is given by \(\displaystyle\int_{0}^{T} f(t) e^{r(T - t)} d t\). Compute the future value after \(6\) years for income received at a rate of \(f(t) = 8000 e^{0.04 t}\) dollars per year and invested at \(6.2 %\) interest.

The present value of an income stream is the amount that would need to be invested now to match the future value as described in Exercise 15 and is given by \(\displaystyle\int_{0}^{T} f(t) e^{-r t} d t\). Find the present value of the income stream in Exercise 15.

Pareto's Law of Income states that the number of people with incomes between \(x = a\) and \(x = b\) is \(N = \displaystyle\int_{a}^{b} A x^{-k} d x\), where \(A\) and \(k\) are constants with \(A > 0\) and \(k > 1\). The average income of these people is \(\bar{x} = \dfrac{1}{N} \displaystyle\int_{a}^{b} A x^{1 - k} d x\). Calculate \(\bar{x}\).

A hot, wet summer is causing a mosquito population explosion in a lake resort area. The number of mosquitoes is increasing at an estimated rate of \(2200 + 10 e^{0.8 t}\) per week (where \(t\) is measured in weeks). By how much does the mosquito population increase between the fifth and ninth weeks of summer?

Use Poiseuille's Law to calculate the rate of flow in a small human artery where we can take \(\eta = 0.027\), \(R = 0.008\) cm, \(l = 2\) cm, and \(P = 4000\) dynes/cm\(^2\).

High blood pressure results from constriction of the arteries. To maintain a normal flow rate (flux), the heart has to pump harder, thus increasing the blood pressure. Use Poiseuille's Law to show that if \(R_0\) and \(P_0\) are normal values of the radius and pressure in an artery and the constricted values are \(R\) and \(P\), then for the flux to remain constant, \(P\) and \(R\) are related by the equation \(\dfrac{P}{P_0} = \left(\dfrac{R_0}{R}\right)^4\). Deduce that if the radius of an artery is reduced to three-fourths of its former value, then the pressure is more than tripled.

The dye dilution method is used to measure cardiac output with \(6\) mg of dye. The dye concentrations, in mg/L, are modeled by \(c(t) = 20 t e^{-0.6 t}\), \(0 \leq t \leq 10\), where \(t\) is measured in seconds. Find the cardiac output.

After a \(5.5\)-mg injection of dye, the readings of dye concentration, in mg/L, at two-second intervals are as shown in the table. Use Simpson's Rule to estimate the cardiac output. \(t\) (s): \(0, 2, 4, 6, 8, 10, 12, 14, 16\) \(c(t)\) (mg/L): \(0.0, 4.1, 8.9, 8.5, 6.7, 4.3, 2.5, 1.2, 0.2\)

The graph of the concentration function \(c(t)\) is shown after a \(7\)-mg injection of dye into a heart. Use Simpson's Rule to estimate the cardiac output.

The demand for a product, in dollars, is \(p = 1200 - 0.2 x - 0.0001 x^2\). Find the consumer surplus when the sales level is \(500\).

A \(5\)-mg bolus of dye is injected into a right atrium. The concentration of the dye (in milligrams per liter) is measured in the aorta at one-second intervals as shown in the table. Estimate the cardiac output. \(t\) (s): \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\) \(c(t)\) (mg/L): \(0, 0.4, 2.8, 6.5, 9.8, 8.9, 6.1, 4.0, 2.3, 1.1, 0\)