Chapter 5

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote. $$ f(x)=\frac{x}{x-3} $$

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: -1,1,3 and \((2, f(2))=(2,4)\)

For the following exercises, use the given length and area of a rectangle to express the width algebraically. Length is \(3 x-4,\) area is \(6 x^{4}-8 x^{3}+9 x^{2}-9 x-4\)

For the following exercises, use the given information about the polynomial graph to write the equation. Double zero at \(x=-3\) and triple zero at \(x=0 .\) Passes through the point (1,32)

For the following exercises, use the written statements to construct a polynomial function that represents the required information. An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of \(d\), the number of days elapsed.

Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote. $$ f(x)=\frac{2 x}{x+4} $$

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: -1,1 (with multiplicity 2 and 1\()\) and \((2, f(2))=(2,4)\)

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. Volume is \(12 x^{3}+20 x^{2}-21 x-36,\) length is \(2 x+3,\) width is \(3 x-4\)

For the following exercises, use the written statements to construct a polynomial function that represents the required information. A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of \(m\), the number of minutes elapsed.

Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote. $$ f(x)=\frac{2 x}{(x-3)^{2}} $$

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: \(-2, \frac{1}{2}\) (with multiplicity 2) and \((-3, f(-3))=(-3,5)\)

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. Volume is \(18 x^{3}-21 x^{2}-40 x+48\), length is \(3 x-4,\) width is \(3 x-4\)

For the following exercises, use the written statements to construct a polynomial function that represents the required information. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by \(x\) inches and the width increased by twice that amount, express the area of the rectangle as a function of \(x\).

Find the dimensions of the rectangular corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote. $$ f(x)=\frac{x^{2}}{x^{2}+2 x+1} $$

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. Volume is \(10 x^{3}+27 x^{2}+2 x-24,\) length is \(5 x-4\) width is \(2 x+3\)

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum. $$ f(x)=x^{4}+x $$

For the following exercises, use the written statements to construct a polynomial function that represents the required information. An open box is to be constructed by cutting out square corners of \(x\) - inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of \(x\)

Among all of the pairs of numbers whose sum is \(6,\) find the pair with the largest product. What is the product?

For the following exercises, use a calculator to graph \(f(x)\). Use the graph to solve \(f(x)>0\). $$ f(x)=\frac{2}{x+1} $$

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: -4,-1,1,4 and \((-2, f(-2))=(-2,10)\)

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. Volume is \(10 x^{3}+30 x^{2}-8 x-24,\) length is \(2,\) width is \(x+3\).

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum. $$ f(x)=-x^{4}+3 x-2 $$

For the following exercises, use the written statements to construct a polynomial function that represents the required information. A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width \((x)\).

Among all of the pairs of numbers whose difference is 12 , find the pair with the smallest product. What is the product?

For the following exercises, use a calculator to graph \(f(x)\). Use the graph to solve \(f(x)>0\). $$ f(x)=\frac{4}{2 x-3} $$

For the following exercises, find the dimensions of the box described. The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically. Volume is \(\pi\left(25 x^{3}-65 x^{2}-29 x-3\right)\), radius is \(5 x+1\).

Suppose that the price per unit in dollars of a cell phone production is modeled by $$p=\$ 45-0.0125 x,$$ where \(x\) is in thousands of phones produced, and the revenue represented by thousands of dollars is $$R=x \cdot p$$. Find the production level that will maximize revenue.

For the following exercises, use a calculator to graph \(f(x)\). Use the graph to solve \(f(x)>0\). $$ f(x)=\frac{2}{(x-1)(x+2)} $$

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically. Volume is \(\pi\left(4 x^{3}+12 x^{2}-15 x-50\right)\), radius is \(2 x+5\)

For the following exercises, find the dimensions of the box described. The length, width, and height are consecutive whole numbers. The volume is 120 cubic inches.

A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by \(h(t)=-4.9 t^{2}+229 t+234\). Find the maximum height the rocket attains.

For the following exercises, use a calculator to graph \(f(x)\). Use the graph to solve \(f(x)>0\). $$ f(x)=\frac{x+2}{(x-1)(x-4)} $$

For the following exercises, find the dimensions of the box described. The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically. Volume is \(\pi\left(3 x^{4}+24 x^{3}+46 x^{2}-16 x-32\right)\), radius is \(x+4\)

A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by \(h(t)=-4.9 t^{2}+24 t+8 .\) How long does it take to reach maximum height?

For the following exercises, use a calculator to graph \(f(x)\). Use the graph to solve \(f(x)>0\). $$ f(x)=\frac{(x+3)^{2}}{(x-1)^{2}(x+1)} $$

For the following exercises, find the dimensions of the box described. The length is three times the height and the height is one inch less than the width. The volume is 108 cubic inches.

A soccer stadium holds 62,000 spectators. With a ticket price of $$\$ 11,$$ the average attendance has been 26,000 . When the price dropped to $$\$ 9,$$ the average attendance rose to 31,000 . Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?

For the following exercises, identify the removable discontinuity. $$ f(x)=\frac{x^{2}-4}{x-2} $$

For the following exercises, find the dimensions of the box described. The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.

For the following exercises, write the polynomial function that models the given situation. A rectangle has a length of 10 units and a width of 8 units. Squares of \(x\) by \(x\) units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of \(x\).

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

For the following exercises, identify the removable discontinuity. $$ f(x)=\frac{x^{3}+1}{x+1} $$

For the following exercises, identify the removable discontinuity. $$ f(x)=\frac{x^{2}+x-6}{x-2} $$

For the following exercises, find the dimensions of the right circular cylinder described. The height is one less than one half the radius. The volume is \(72 \pi\) cubic meters.

For the following exercises, write the polynomial function that models the given situation. A square has sides of 12 units. Squares \(x+1\) by \(x+1\) units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of \(x\).