Chapter 10

Find the dimensions of a rectangle of area 144 sq \(\mathrm{ft}\) that has the smallest possible perimeter.

The owner of the Rancho Los Feliz has 3000 yd of fencing with which to enclose a rectangular piece of grazing land along the straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area that he can enclose? What is this area?

The management of the UNICO department store has decided to enclose an \(800-\mathrm{ft}^{2}\) area outside the building for displaying potted plants and flowers. One side will be formed by the external wall of the store, two sides will be constructed of pine boards, and the fourth side will be made of galvanized steel fencing. If the pine board fencing costs $$\$ 6 /$$ running foot and the steel fencing costs $$\$ 3 /$$ running foot, determine the dimensions of the enclosure that can be erected at minimum cost.

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 15 in. long and 8 in. wide, find the dimensions of the box that will yield the maximum volume.

If an open box is made from a tin sheet 8 in. square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made.

If an open box has a square base and a volume of \(108 \mathrm{in} .{ }^{3}\) and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction.

What are the dimensions of a closed rectangular box that has a square cross section, a capacity of 128 in. \(^{3}\), and is constructed using the least amount of material?

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=-x^{2}+4 x+3 $$

Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 108 in. Find the dimensions of a rectangular package that has a square cross section and the largest volume that may be sent via priority mail. What is the volume of such a package? Hint: The length plus the girth is \(4 x+h\) (see the accompanying figure).

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=\frac{1}{x} $$

A book designer has decided that the pages of a book should have 1 -in. margins at the top and bottom and \(\frac{1}{2}\) -in. margins on the sides. She further stipulated that each page should have an area of \(50 \mathrm{in} .{ }^{2}\) (see the accompanying figure). Determine the page dimensions that will result in the maximum printed area on the page.

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=\frac{1}{x+2} $$

Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 108 in. Find the dimensions of the cylindrical package of greatest volume that may be sent via priority mail. What is the volume of such a package? Compare with Exercise \(11 .\) Hint: The length plus the girth is \(2 \pi r+l\).

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=\frac{1}{1+x^{2}} $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=-\frac{2}{x^{2}} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=3 x+5 $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=\frac{x}{1+x^{2}} $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ g(x)=\frac{1}{1+2 x^{2}} $$

Determine which graph-(a), (b), or (c)-is the graph of the function \(f\) with the specified properties. \(f\) is decreasing on \((-\infty, 2)\) and increasing on \((2, \infty), f\) is concave upward on \((1, \infty)\), and \(f\) has inflection points at \(x=0\) and \(x=1\).

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=4-5 x $$

The cabinet that will enclose the Acrosonic model D loudspeaker system will be rectangular and will have an internal volume of \(2.4 \mathrm{ft}^{3}\). For aesthetic reasons, it has been decided that the height of the cabinet is to be \(1.5\) times its width. If the top, bottom, and sides of the cabinet are constructed of veneer costing \(40 \phi /\) square foot and the front (ignore the cutouts in the baffle) and rear are constructed of particle board costing \(20 \phi /\) square foot, what are the dimensions of the enclosure that can be constructed at a minimum cost?

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=x^{2}-2 x-3 \text { on }[-2,3] $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=\frac{x-1}{x+1} $$

The following graphs were used by the CEO of the Madison Savings Bank to illustrate what effect a projected promotional campaign would have on its deposits over the next year. The functions \(D_{1}\) and \(D_{2}\) give the projected amount of money on deposit with the bank over the next 12 mo with and without the proposed promotional campaign, respectively. a. Determine the signs of \(D_{1}^{\prime}(t), D_{2}^{\prime}(t), D_{1}^{\prime \prime}(t)\), and \(D_{2}^{\prime \prime}(t)\) on the interval \((0,12)\). b. What can you conclude about the rate of change of the growth rate of the money on deposit with the bank with and without the proposed promotional campaign?

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=x^{2}-3 x $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=x^{2}-2 x-3 \text { on }[0,4] $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ g(t)=\frac{t+1}{2 t-1} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=2 x^{2}+x+1 $$

If exactly 200 people sign up for a charter flight, Leisure World Travel Agency charges $$\$300$$/person. However, if more than 200 people sign up for the flight (assume this is the case), then each fare is reduced by $$\$ 1$$ for each additional person. Determine how many passengers will result in a maximum revenue for the travel agency. What is the maximum revenue? What would be the fare per passenger in this case? Hint: Let \(x\) denote the number of passengers above 200 . Show that the revenue function \(R\) is given by \(R(x)=(200+x)(300-x)\).

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=-x^{2}+4 x+6 \text { on }[0,5] $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ g(x)=x-x^{3} $$

An apple orchard has an average yield of 36 bushels of apples/tree if tree density is 22 trees/acre. For each unit increase in tree density, the yield decreases by 2 bushels/tree. How many trees should be planted in order to maximize the yield?

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=-x^{2}+4 x+6 \text { on }[3,6] $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=x^{3}-3 x^{2} $$

The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $$\$ 600 /$$ person/ day if exactly 20 people sign up for the cruise. However, if more than 20 people sign up (up to the maximum capacity of 90 ) for the cruise, then each fare is reduced by $$\$ 4$$ for each additional passenger. Assuming at least 20 people sign up for the cruise, determine how many passengers will result in the maximum revenue for the owner of the yacht. What is the maximum revenue? What would be the fare/passenger in this case?

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=x^{3}+3 x^{2}-1 \text { on }[-3,2] $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(t)=\frac{t^{2}}{t^{2}-9} $$

Show that the function is concave upward wherever it is defined. $$ f(x)=4 x^{2}-12 x+7 $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ g(x)=x^{3}+3 x^{2}+1 $$

Phillip, the proprietor of a vineyard, estimates that the first 10,000 bottles of wine produced this season will fetch a profit of $$\$ 5 /$$ bottle. But if more than 10,000 bottles were produced, then the profit/bottle for the entire lot would drop by $$\$ 0.0002$$ for each additional bottle sold. Assuming at least 10,000 bottles of wine are produced and sold, what is the maximum profit?

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=x^{3}+3 x^{2}-1 \text { on }[-3,1] $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ g(x)=\frac{x^{3}}{x^{2}-4} $$

Show that the function is concave upward wherever it is defined. $$ g(x)=x^{4}+\frac{1}{2} x^{2}+6 x+10 $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=x^{3}-3 x+4 $$

OPTIMAL SPEED OF A TruCK A truck gets \(600 / x\) mpg when driven at a constant speed of \(x\) mph (between 50 and \(70 \mathrm{mph}\) ). If the price of fuel is $$\$ 3$$ /gallon and the driver is paid $$\$ 18 /$$ hour, at what speed between 50 and \(70 \mathrm{mph}\) is it most economical to drive?

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=3 x^{4}+4 x^{3} \text { on }[-2,1] $$

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=\frac{3 x}{x^{2}-x-6} $$

Show that the function is concave upward wherever it is defined. $$ f(x)=\frac{1}{x^{4}} $$

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=\frac{1}{3} x^{3}-3 x^{2}+9 x+20 $$

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=\frac{1}{2} x^{4}-\frac{2}{3} x^{3}-2 x^{2}+3 \text { on }[-2,3] $$