Chapter 5

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

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.

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 \(2 x+5\)

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.

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, 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 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, 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 is iven by \(h(t)=-4.9 t^{2}+24 t+8 .\) How long does it take to reach maximum height?

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.

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.

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

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?

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

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?

A farmer fi ds 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, find the dimensions of the right circular cylinder described. The radius is 3 inches more than the height. The volume is 16\(\pi\) cubic meters.

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

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

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 \(x .\)

For the following exercises, find the dimensions of the right circular cylinder described. The radius and height differ by one meter. The radius is larger and the volume is 48\(\pi\) cubic meters.

For the following exercises, identify the removable discontinuity. $$ f(x)=\frac{2 x^{2}+5 x-3}{x+3} $$

For the following exercises, write the polynomial function that models the given situation. A cylinder has a radius of \(x+2\) units and a height of 3 units greater. Express the volume of the cylinder as a polynomial function.

For the following exercises, find the dimensions of the right circular cylinder described. The radius and height differ by two meters. The height is greater and the volume is 28.125\(\pi\) cubic meters.

For the following exercises, identify the removable discontinuity. $$f(x)=\frac{x^{3}+x^{2}}{x+1}$$

For the following exercises, write the polynomial function that models the given situation. A right circular cone has a radius of \(3 x+6\) and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is \(V=\frac{1}{3} \pi r^{2} h\) for radius \(r\) and height \(h .\)

For the following exercises, find the dimensions of the right circular cylinder described. The radius is \(\frac{1}{3}\) meter greater than the height. The volume is \(\frac{98}{9 \pi} \pi\) cubic meters.

For the following exercises, express a rational function that describes the situation. A large mixing tank currently contains 200 gallons of water, into which 10 pounds of sugar have been mixed. A tap will open, pouring 10 gallons of water per minute into the tank at the same time sugar is poured into tank at a rate of 3 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after \(t\) minutes.

For the following exercises, express a rational function that describes the situation. A large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after \(t\) minutes.

For the following exercises, use the given rational function to answer the question. The concentration \(C\) of a drug in a patient's bloodstream \(t\) hours after injection in given by \(C(t)=\frac{2 t}{3+t^{2}} .\) What happens to the concentration of the drug as \(t\) increases?

For the following exercises, use the given rational function to answer the question. The concentration \(C\) of a drug in a patient's bloodstream \(t\) hours after injection is given by \(C(t)=\frac{100 t}{2 t^{2}+75} .\) Use a calculator to approximate the time when the concentration is highest.

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question. An open box with a square base is to have a volume of 108 cubic inches. Find the dimensions of the box that will have minimum surface area. Let \(x=\) length of the side of the base.

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question. A rectangular box with a square base is to have a volume of 20 cubic feet. The material for the base costs 30 cents/square foot. The material for the sides costs 10 cents/square foot. The material for the top costs 20 cents/square foot. Determine the dimensions that will yield minimum cost. Let \(x=\) length of the side of the base.

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question. A right circular cylinder with no top has a volume of 50 cubic meters. Find the radius that will yield minimum surface area. Let \(x=\) radius.

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question. A right circular cylinder is to have a volume of 40 cubic inches. It costs 4 cents/square inch to construct the top and bottom and 1 cent/square inch to construct the rest of the cylinder. Find the radius to yield minimum cost. Let \(x=\) radius.