91Ó°ÊÓ

Find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.) $$ \begin{array}{l}{\text { Plane: } x+y+z=1,(2,1,1)} \\ {\text { Minimize } d^{2}=(x-2)^{2}+(y-1)^{2}+(z-1)^{2}}\end{array} $$

Find the standard equation of the sphere. $$ \text { Endpoints of a diameter: }(2,0,0),(0,6,0) $$

Production The production function for a company is given by \(f(x, y)=100 x^{0.25} y^{0.75}\) where \(x\) is the number of units of labor and \(y\) is the number of units of capital. Suppose that labor costs dollar 48 per unit and capital costs dollar 36 per unit. The total cost of labor and capital is limited to dollar 100,000 (a) Find the maximum production level for this manufacturer. (b) Find the marginal productivity of money. (c) Use the marginal productivity of money to find the maximum number of units that can be produced if dollar 125,000 is available for labor and capital.

Profit A corporation manufactures candles at two locations. The cost of producing \(x_{1}\) units at location 1 is \(C_{1}=0.02 x_{1}^{2}+4 x_{1}+500\) and the cost of producing \(x_{2}\) units at location 2 is \(C_{2}=0.05 x_{2}^{2}+4 x_{2}+275\) The candles sell for 15 dollars per unit. Find the quantity that should be produced at each location to maximize the profit \(P=15\left(x_{1}+x_{2}\right)-C_{1}-C_{2}\).

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ 2 x-3 y=0, x+y=5, y=0 $$

Identify the quadric surface. $$ x^{2}-y^{2}+z=0 $$

Hardy-Weinberg Law Common blood types are determined genetically by the three alleles A, B, and O. (An allele is any of a group of possible mutational forms of a gene.) A person whose blood type is AA, BB, or OO is homozygous. A person whose blood type is AB, AO, or BO is heterozygous. The Hardy-Weinberg Law states that the proportion \(P\) of heterozygous individuals in any given population is modeled by \(P(p, q, r)=2 p q+2 p r+2 q r\) where \(p\) represents the percent of allele \(\mathrm{A}\) in the population, \(q\) represents the percent of allele \(\mathrm{B}\) in the population, and \(r\) represents the percent of allele \(\mathrm{O}\) in the population. Use the fact that \(p+q+r=1\) (the sum of the three must equal \(100 \%\)) to show that the maximum proportion of heterozygous individuals in any population is \(\frac{2}{3} .\)

Sketch the \(x y\) -trace of the sphere. $$ x^{2}+y^{2}+z^{2}-6 x-10 y+6 z+30=0 $$

Identify the quadric surface. $$ z^{2}=2 x^{2}+2 y^{2} $$

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=\frac{x}{x+y} $$