Chapter 1

The quantity demanded \(x\) (measured in units of a thousand) of a certain commodity when the unit price is set at \(\$ p\) is given by the equation $$ p=\sqrt{-x^{2}+100} $$ If the unit price is set at \(\$ 6\), what is the quantity demanded?

Write the expression in simplest radical form. $$ \sqrt[3]{\frac{3 a^{3}}{b^{2}}} $$

The quantity \(x\) of satellite radios that a manufacturer will make available in the marketplace is related to the unit price \(p\) (in dollars) by the equation $$ p=\frac{1}{10} \sqrt{x}+10 $$ How many satellite radios will the manufacturer make available in the marketplace if the unit price is \(\$ 30\) ?

Simplify the expression. $$ \frac{1}{\sqrt{a}}+\sqrt{a} $$

Simplify the expression. $$ \frac{x}{\sqrt{x-y}}-\sqrt{x-y} $$

When organic waste is dumped into a pond, the oxidation process that takes place reduces the pond's oxygen content. However, given time, nature will restore the oxygen content to its natural level. Suppose the oxygen content \(t\) days after organic waste has been dumped into the pond is given by $$ P=100\left(\frac{t^{2}+10 t+100}{t^{2}+20 t+100}\right) $$ percent of its normal level. Find \(t\) corresponding to an oxygen content of \(80 \%\) and interpret your results.

Simplify the expression. $$ \frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}}+\frac{\sqrt{y}}{\sqrt{x}-\sqrt{y}} $$

Consider a rectangle of width \(x\) and height \(y\) (see the accompanying figure). The ratio \(r=\frac{x}{y}\) satisfying the equation $$ \frac{x}{y}=\frac{x+y}{x} $$ is called the golden ratio. Show that $$ r=\left(\frac{1}{2}\right)(1+\sqrt{5}) \approx 1.6 $$.

Simplify the expression. $$ \frac{a}{\sqrt{a^{2}-b^{2}}}-\frac{\sqrt{a^{2}-b^{2}}}{a} $$

A Box 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 (see the accompanying figure). If the cardboard is 16 in. long and 10 in. wide, find the dimensions of the resulting box if it is to have a total surface area of \(144 \mathrm{in}\). \(^{2}\)

Simplify the expression. $$ (x+1)^{1 / 2}+\frac{1}{2} x(x+1)^{-1 / 2} $$

Carmen wishes to put up a fence around a proposed rectangular garden in her backyard. The length of the garden is to be twice its width, and the area of the garden is to be \(200 \mathrm{ft}^{2}\). How many feet of fencing does she need?

Simplify the expression. $$ \frac{1}{2} x^{-1 / 2}(x+y)^{1 / 3}+\frac{1}{3} x^{1 / 2}(x+y)^{-2 / 3} $$

George has \(120 \mathrm{ft}\) of fencing. He wishes to cut it into two pieces, with the purpose of enclosing two square regions. If the sum of the areas of the regions enclosed is \(562.5 \mathrm{ft}^{2}\), how long should each piece of fencing be?

Simplify the expression. $$ \frac{\frac{1}{2}\left(1+x^{1 / 3}\right) x^{-1 / 2}-\frac{1}{3} x^{1 / 2} \cdot x^{-2 / 3}}{\left(1+x^{1 / 3}\right)^{2}} $$

A rectangular garden of length \(40 \mathrm{ft}\) and width \(20 \mathrm{ft}\) is surrounded by a path of uniform width. If the area of the walkway is \(325 \mathrm{ft}^{2}\), what is its width?

Simplify the expression. $$ \frac{\frac{1}{2} x^{-1 / 2}(x+y)^{1 / 2}-\frac{1}{2} x^{1 / 2}(x+y)^{-1 / 2}}{x+y} $$

The owner of the Rancho los Feliz has 3000 yd of fencing to enclose a rectangular piece of grazing land along the straight portion of a river. What are the dimensions of the largest area that can be enclosed?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(a\) is a real number, then \(\sqrt{a^{2}}=|a|\).

RADIUS OF A CrLINDRICAL CAN The surface area of a right circular cylinder is given by \(S=2 \pi r^{2}+2 \pi r h\), where \(r\) is the radius of the cylinder and \(h\) is its height. What is the radius of a cylinder of surface area \(100 \mathrm{in} .{ }^{2}\) and height 3 in.?

A metal container consists of a right circular cylinder with hemispherical ends. The surface area of the container is \(S=2 \pi r l+4 \pi r^{2}\), where \(l\) is the length of the cylinder and \(r\) is the radius of the hemisphere. If the length of the cylinder is \(4 \mathrm{ft}\) and the surface area of the container is \(28 \pi \mathrm{ft}^{2}\), what is the radius of each hemisphere?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(n\) is a natural number and \(a\) is a positive real number, then \(\left(a^{1 / n}\right)^{n}=a\)

In calm waters the oil spilling from the ruptured hull of a grounded oil tanker spreads in all directions. The area polluted at a certain instant of time was circular with a radius of \(100 \mathrm{ft}\). A little later, the area, still circular, had increased by \(4400 \pi \mathrm{ft}^{2}\). By how much had the radius increased?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(a\) and \(b\) are real numbers and \(a b \neq 0\), then \(a \neq 0\) or \(b \neq 0 .\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(b^{2}-4 a c>0\) and \(a \neq 0\), then the roots of \(a x^{2}-b x+\) \(c=0\) are the negatives of the roots of \(a x^{2}+b x+c=0 .\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(b^{2}-4 a c \neq 0\) and \(a \neq 0\), then \(a x^{2}+b x+c=0\) has two distinct real roots, or it has no real roots at all.