Chapter 12

A centrifuge is a machine designed for the specific purpose of subjecting materials to a sustained centrifugal force. The actual amount of centrifugal force, \(F\), expressed in dynes ( 1 gram of force \(=980\) dynes) is given by $$F=f(M, S, R)=\frac{\pi^{2} S^{2} M R}{900}$$ where \(S\) is in revolutions per minute (rpm), \(M\) is in grams, and \(R\) is in centimeters. Show that an object revolving at the rate of \(600 \mathrm{rpm}\) in a circle with radius of \(10 \mathrm{~cm}\) generates a centrifugal force that is approximately 40 times gravity.

The productivity of a country in Western Europe is given by the function $$f(x, y)=40 x^{4 / 5} y^{1 / 5}$$ when \(x\) units of labor and \(y\) units of capital are used. a. What is the marginal productivity of labor and the marginal productivity of capital when the amounts expended on labor and capital are 32 units and 243 units, respectively? b. Should the government encourage capital investment rather than increased expenditure on labor at this time in order to increase the country's productivity?

LAND PRICES The rectangular region \(R\) shown in the following figure represents a city's financial district. The price of land within the district is approximated by the function $$p(x, y)=200-10\left(x-\frac{1}{2}\right)^{2}-15(y-1)^{2}$$ where \(p(x, y)\) is the price of land at the point \((x, y)\) in dollars per square foot and \(x\) and \(y\) are measured in miles. Compute $$\frac{\partial p}{\partial x}(0,1) \text { and } \frac{\partial p}{\partial y}(0,1)$$ and interpret your results.

According to the ideal gas law, the volume \(V\) of an ideal gas is related to its pressure \(P\) and temperature \(T\) by the formula $$V=\frac{k T}{P}$$ where \(k\) is a positive constant. Describe the level curves of \(V\) and give a physical interpretation of your result.

Drafted by an international committee in 1989 , the rules for the new International America's Cup Class (IACC) include a formula that governs the basic yacht dimensions. The formula $$f(L, S, D) \leq 42$$ where $$f(L, S, D)=\frac{L+1.25 S^{1 / 2}-9.80 D^{1 / 3}}{0.388}$$ balances the rated length \(L\) (in meters), the rated sail area \(S\) (in square meters), and the displacement \(D\) (in cubic meters). All changes in the basic dimensions are trade-offs. For example, if you want to pick up speed by increasing the sail area, you must pay for it by decreasing the length or increasing the displacement, both of which slow down the boat. Show that yacht A of rated length \(20.95 \mathrm{~m}\), rated sail area \(277.3 \mathrm{~m}^{2}\), and displacement \(17.56 \mathrm{~m}^{3}\) and the longer and heavier yacht \(\mathrm{B}\) with \(L=21.87, S=311.78\), and \(D=\) \(22.48\) both satisfy the formula.

In a survey it was determined that the demand equation for VCRs is given by $$x=f(p, q)=10,000-10 p-e^{0.5 q}$$ The demand equation for blank VCR tapes is given by $$y=g(p, q)=50,000-4000 q-10 p$$ where \(p\) and \(q\) denote the unit prices, respectively, and \(x\) and \(y\) denote the number of VCRs and the number of blank VCR tapes demanded each week. Determine whether these two products are substitute, complementary, or neither.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(h\) is a function of \(x\) and \(y\), then there are functions \(f\) and \(g\) of one variable such that $$h(x, y)=f(x)+g(y)$$

The total weekly revenue (in dollars) of Country Workshop associated with manufacturing and selling their rolltop desks is given by the function \(R(x, y)=-0.2 x^{2}-0.25 y^{2}-0.2 x y+200 x+160 y\) where \(x\) denotes the number of finished units and \(y\) denotes the number of unfinished units manufactured and sold each week. Compute \(\partial R / \partial x\) and \(\partial R / \partial y\) when \(x=300\) and \(y=250\). Interpret your results.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The domain of \(f(x, y)=1 /\left(x^{2}-y^{2}\right)\) is \(\\{(x, y) \mid y \neq x\\}\).

The monthly profit (in dollars) of Bond and Barker Department Store depends on the level of inventory \(x\) (in thousands of dollars) and the floor space \(y\) (in thousands of square feet) available for display of the merchandise, as given by the equation $$\begin{aligned}P(x, y)=&-0.02 x^{2}-15 y^{2}+x y \\ &+39 x+25 y-20,000\end{aligned}$$ Compute \(\partial P / \partial x\) and \(\partial P / \partial y\) when \(x=4000\) and \(y=150\). Interpret your results. Repeat with \(x=5000\) and \(y=150\).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Every point on the level curve \(f(x, y)=c\) corresponds to a point on the graph of \(f\) that is \(c\) units above the \(x y\) -plane if \(c>0\) and \(|c|\) units below the \(x y\) -plane if \(c<0 .\)

A formula used by meteorologists to calculate the wind chill temperature (the temperature that you feel in still air that is the same as the actual temperature when the presence of wind is taken into consideration) is\(T=f(t, s)=35.74+0.6215 t-35.75 s^{0.16}+0.4275 t s^{0.16}$$(s \geq 1)\) where \(t\) is the actual air temperature in degrees Fahrenheit and \(s\) is the wind speed in mph. a. What is the wind chill temperature when the actual air temperature is \(32^{\circ} \mathrm{F}\) and the wind speed is \(20 \mathrm{mph}\) ? b. If the temperature is \(32^{\circ} \mathrm{F}\), by how much approximately will the wind chill temperature change if the wind speed increases from \(20 \mathrm{mph}\) to \(21 \mathrm{mph}\) ?

The efficiency of an internal combustion engine is given by $$E=\left(1-\frac{v}{V}\right)^{04}$$ where \(V\) and \(v\) are the respective maximum and minimum volumes of air in each cylinder. a. Show that \(\partial E / \partial V>0\) and interpret your result. b. Show that \(\partial E / \partial v<0\) and interpret your result.

The volume \(V\) (in liters) of a certain mass of gas is related to its pressure \(P\) (in millimeters of mercury) and its temperature \(T\) (in degrees Kelvin) by the law $$V=\frac{30.9 T}{P}$$ Compute \(\partial V / \partial T\) and \(\partial V / \partial P\) when \(T=300\) and \(P=800\). Interpret your results.

The formula $$S=0.007184 W^{0.425} H^{0.725}$$ gives the surface area \(S\) of a human body (in square meters) in terms of its weight \(W\) (in kilograms) and its height \(H\) (in centimeters). Compute \(\partial S / \partial W\) and \(\partial S / \partial H\) when \(W=70 \mathrm{~kg}\) and \(H=180 \mathrm{~cm}\). Interpret your results.

According to the ideal gas law, the volume \(V\) (in liters) of an ideal gas is related to its pressure \(P\) (in pascals) and temperature \(T\) (in degrees Kelvin) by the formula $$V=\frac{k T}{P}$$ where \(k\) is a constant. Show that $$\frac{\partial V}{\partial T} \cdot \frac{\partial T}{\partial P} \cdot \frac{\partial P}{\partial V}=-1$$

The kinetic energy \(K\) of a body of mass \(m\) and velocity \(v\) is given by $$K=\frac{1}{2} m v^{2}$$ Show that \(\frac{\partial K}{\partial m} \cdot \frac{\partial^{2} K}{\partial v^{2}}=K\).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f_{x y}(x, y)\) and \(f_{y x}(x, y)\) are both continuous for all values of \(x\) and \(y\), then \(f_{x y}=f_{y x}\) for all values of \(x\) and \(y\).