91Ó°ÊÓ

Evaluate: a. |9| b. |-9| c. |-1000| d. \(-|-1000|\)

Without using a calculator show how you can solve for \(x\). a. \(10^{x-5}=1000\) b. \(\log (2 x+10)=2\) c. \(10^{3 x-1}=0.0001\) d. \(\log (500-25 x)=3\)

a. In 2006 Japan had a population of approximately 127.5 million people and a total land area of about 152.5 thousand square miles. What was the population density (the number of people per square mile)? b. In 2006 the United States had a population of approximately 300 million people and a total land area of about 3620 thousand square miles. What was the population density of the United States? c. Compare the population densities of Japan and the United States.

Determine the value of each expression. a. \(|-5-3|\) c. \(|2-6|\) b. \(|6-2|\) d. \(-2|-1+3|+|-5|\)

In the United States, land is measured in acres and one acre is 43,560 sq \(\mathrm{ft}\) a. If you buy a one-acre lot that is in the shape of a square, what would be the length of each side in feet? b. A newspaper advertisement states that all lots in a new housing development will be a minimum of one and a half acres. Assuming the lot is rectangular and has \(150 \mathrm{ft}\) of frontage, how deep will the minimal-size lot be? If the new home owner wants to fence in the lot, how many yards of fencing would be needed? c. The metric unit for measuring land is the square hectometer. (A hectometer is a length of 100 meters.) Find the size of a one-acre lot if it were measured in square hectometers. d. A hectare is 100 acres. How many one-acre lots can fit in a square mile? How many hectares is that?

Change each number into scientific notation, then perform the indicated calculation without a calculator. a. A \(\$ 600,000\) lottery jackpot is divided among 300 people. What are the winnings per person? b. A total of 2500 megawatts are used over 500 hours. What is the rate in watts per hour? c. If there were 6 million births in 30 years, what is the birth rate per year?

A nanosecond is \(10^{-9}\) second. Modern computers can perform on the order of one operation every nanosecond. Approximately how many feet does an electrical signal moving at the speed of light travel in a computer in 1 nanosecond?

a. For any nonzero real number \(a\), what can we say about the sign of the expression \((-a)^{n}\) when \(n\) is an even integer? What can we say about the sign of \((-a)^{n}\) when \(n\) is an odd integer? b. What is the sign of the resulting number if \(a\) is a positive number? If \(a\) is a negative number? Explain your answer.

Radio waves, sent from a broadcast station and picked up by the antenna of your radio, are a form of electromagnetic (EM) radiation, as are microwaves, X-rays, and visible, infrared, and ultraviolet light. They all travel at the speed of light. Electromagnetic radiation can be thought of as oscillations like the vibrating strings of a violin or guitar or like ocean swells that have crests and troughs. The distance between the crest or peak of one wave and the next is called the wavelength. The number of times a wave crests per minute, or per second for fast-oscillating waves, is called its frequency. Wavelength and frequency are inversely proportional: the longer the wavelength, the lower the frequency, and vice versa-the faster the oscillation, the shorter the wavelength. For radio waves and other \(\mathrm{EM}\), the number of oscillations per second of a wave is measured in hertz, after the German scientist who first demonstrated that electrical waves could transmit information across space. One cycle or oscillation per second equals 1 hertz \((\mathrm{Hz})\). For the following exercise you may want to find an old radio or look on a stereo tuner at the AM and FM radio bands. You may see the notation \(\mathrm{kHz}\) beside the AM band and MHz beside the FM band. AM radio waves oscillate at frequencies measured in the kilohertz range, and FM radio waves oscillate at frequencies measured in the megahertz range. a. The Boston FM rock station WBCN transmits at \(104.1 \mathrm{MHz}\). Write its frequency in hertz using scientific notation. b. The Boston AM radio news station WBZ broadcasts at 1030 \(\mathrm{kHz}\). Write its frequency in hertz using scientific notation. The wavelength \(\lambda\) (Greek lambda) in meters and frequency \(\mu\) (Greek mu) in oscillations per second are related by the formula \(\lambda=\frac{c}{\mu}\) where \(c\) is the speed of light in meters per second. c. Estimate the wavelength of the WBCN FM radio transmission. d. Estimate the wavelength of the WBZ AM radio transmission. e. Compare your answers in parts (c) and (d), using orders of magnitude, with the length of a football field (approximately 100 meters).

An equilateral triangle has sides of length \(8 \mathrm{~cm}\). a. Find the height of the triangle. (Hint: Use the Pythagorean theorem on the inside back cover.) b. Find the area \(A\) of the triangle if \(A=\frac{1}{2} b h\).