Chapter 5

For the following exercises, determine the function described and then use it to answer the question. Consider a cone with height of 30 feet. Express the radius, \(r\) , in terms of the volume, \(V\) , and find the radius of a cone with volume of 1000 cubic feet.

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. \(f(x)=16 x^{4}-24 x^{3}+x^{2}-15 x+25\)

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

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

For the following exercises, use the given length and area of a rectangle to express the width algebraically. Length is \(2 x+5,\) area is \(4 x^{3}+10 x^{2}+6 x+15\)

For the following exercises, use the given information about the polynomial graph to write the equation. Degree \(4 .\) Roots of multiplicity 2 at \(x=\frac{1}{2}\) and roots of multiplicity 1 at \(x=6\) and \(x=-2 . y\) -intercept at \((0,18) .\)

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains \((1,-6)\) has the shape of \(f(x)=3 x^{2}\) Vertex has \(x\) -coordinate of \(-1 .\)

Use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or ?1. There may be more than one correct answer. The \(y\) -intercept is \((0,1)\) . There is no \(x\) -intercept. Degree is \(4 .\) End behavior: as \(x \rightarrow-\infty, f(x) \rightarrow \infty,\) as \(x \rightarrow \infty, f(x) \rightarrow \infty.\)

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

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.

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

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, 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 \((1,32)\) .

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

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.

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, make tables to behavior of the function near the vertical asymptote and reffecting the horizontal asymptote $$ f(x)=\frac{2 x}{x+4} $$

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

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.

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 a calculator to approximate local minima and maxima or the global minimum and maximum. $$ f(x)=x^{3}-x-1 $$

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.

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.

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 \((-3, f(-3))=(-3,5)\)

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 \(x\).

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, 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 \(3 x-4\)

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.

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 \(x .\)

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum. $$ f(x)=2 x^{3}-3 x-1 $$

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 \(x\)

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. $$ f(x)=x^{4}+x $$

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

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 \(x .\)

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

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 \((x)\).

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, 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. $$ f(x)=-x^{4}+3 x-2 $$

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

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 \((x).\)

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, 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 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, 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 \(5 x+1\)

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum. $$ f(x)=x^{4}-x^{3}+1 $$

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