Math Review

Mathematically, what is the correct way to write a speed of 100 meters per second?

100 m/s

The frequency of a photon of light is related to its wavelength by the following formula: frequency= c/wavelength where c=3×10^8 m/s is the speed of light. What is the frequency of a photon with a wavelength of 5×10^−9 meter?

6 x 10^16 /s

(10^3 X 10^2) + 10^-3=

100,000.001

17^3×17^−7=

17^-4

2×10^7/8×10^5=

2.5×10^1

There are 3 feet in 1 yard. Therefore, 1 square yard is equivalent to _____.

9 ft^2

Consider the types of units that go with the following equations. Which statement is valid?

distance=speed×time

3/8=

0.375

The average distance from Earth to the Moon is 384,400 kilometers. What is this distance in miles, rounded to the nearest 100 miles?

238,900 miles

(1.5×10^−7)×(3.6×10^15)÷(1.2×10^−3)=

4.5×10^11

8√ is between _____.

2 and 3

The surface area of a small moon is 50,000 square kilometers, which is equivalent to _____ square meters. (Recall that 1 kilometer = 1000 meters.)

50,000×1000^2

A typical galaxy contains about 100 billion stars, and there are approximately 100 billion galaxies in the observable universe. About how many stars are in the observable universe?

10^22 Correct We can write the calculation formally like this: 10^11 stars/galaxy ×10^11 galaxy=10^22 stars Note how the units of "galaxy" (or "galaxies") cancel, leaving the answer of 10^22 stars. This number is easy to write, but it is incredibly large. In fact, the number of stars in the observable universe is comparable to the number of grains of dry sand on all the beaches on Earth combined.

You drop a rock and it accelerates with the acceleration of gravity, 9.8 m/s^2. How fast will it be going after 2 seconds?

19.6 m/s

The table below reviews the basic rules for deciding when digits are significant. Use these rules to complete Part A. Before you begin, you may wish to watch the video Precision and Significant Digits, Part 1.

A measurement of 200,000 centimeters has 1 significant digit(s). A measurement of 2.00000 x 10^5 centimeters has 6 significant digit(s). A measurement of 700 seconds has 1 significant digit(s). A measurement of 700.1 seconds has 4 significant digit(s). A measurement of 0.0005 microgram has 1 significant digit(s). A measurement of 1.0005 has 5 significant digit(s).

Part D - Estimating Roots Use the labels on the left to complete the estimation sentences on the right. The video Powers and Roots, Part 2, reviews the ideas you'll need for this Part. Drag the appropriate labels to their respective targets. Note: You may use the same label more than once, and not all labels will be used.

The square root of 2 is between 1 and 2. The square root of 20 is between 4 and 5. The square root of 37 is between 6 and 7. The square root of 5 is between 2 and 3. 3 square root 26 is between 2 and 3. 4 square root 17 is between 2 and 3. 3 square root 1200 is between 10 and 11.

10^-7

0.0000001

The width of your thumb is about _____.

2 centimeters

If you assume that there are exactly 365 days in a year, how many seconds are there in one year? Give your answer to the nearest 1000 seconds.

31,536,000 seconds

Scientific notation 4.2×10^−2 / 8.4×10^−5=

5 x 10^2

In scientific notation, 7094.2=

7.0942 x 10^3

What is the mass of a typical adult?

75 kilograms

How many significant digits are in the number 14,050.010?

8

10^4 / 10^-1= / is a divided by sign

100,000

10^5+10^−1=

100,000.1

Recall that 1 centimeter = 10 millimeters. Therefore, 1 cubic centimeter = _____ cubic millimeters.

1000

(10^4)^3÷10^2=

10^10

At a supermarket in France, the price of apples is 2.50 euros per kilogram. Suppose the exchange rate is 1 euro = $1.35. What is the price of the apples in dollars per pound? Recall that 1 kilogram=2.205 pounds.

$1.53/pound

Part B - Rules for Working with Powers of 10 Identify the equations below that are true. You may wish to watch the video Powers of 10, Part 3 for a review of operations (adding, subtracting, multiplying, dividing) with powers of 10. Select all that apply.

(10^5)^3=10^5×3 10^5÷10^3=10^5−3 10^5×10^3=10^5+3 10^5/10^3=10^5−3

(10^2 X 10^3)^-2 =

0.0000000001

The Moon's average density is about 3.34 grams per cubic centimeter. What is this density in units of pounds per cubic inch?

0.121 pounds per cubic inch

Part B - Comparing Metric Units The table below shows the metric prefixes. Use this table to answer the following questions. Before you begin, you may wish to watch the video Metric Units, Part 1.

1 kilometer is 10^6 times as large as 1 millimeter. 1 hectometer is 10^4 times as large as centimeter. 1 centimeter is 10^4 times as large as 1 micrometer. 1 millimeter is 10^6 times as large as 1 nanometer. 1 megameter is 10^3 times as large as 1 kilometer. 1 gigameter is 10^6 times as large as 1 kilometer.

Part E - Power Rules Identify the equations below that are true. The video Powers and Roots, Part 3, reviews the ideas you'll need for this part. Select all that apply.

1. 3^4 X 3^5 = 3^4+5 2. 3^4 divided by sign 3^5 = 3^4-5 3. 9^-3 X 9^-6 = 9^-3+(-6) 4. Square root 2 X square root 2 = 2 1/2+1/2 5. 12^2/12^3 = 12^2-3 6. (7^-2)^3 = 7^-2X3

Part A - Common Fractions Use the fractions at left to complete the statements at right. The video Fractions, Part 1 reviews the ideas used in this part. Drag the appropriate labels to their respective targets. Note: You may use the same label more than once, and not all labels will be used.

1. The fraction 2/3 has a numerator of 2 and a denominator of 3. 2. -12 can be written as the common fraction -12/1. 3. Dividing 2 pieces of pie into 3 pieces represents the fraction 2/3. 4. The reciprocal of 2/3 is 3/2. 5. 5 candy bars are divided among 6 people. Each person gets 5/6 candy bar(s). 6. To add 1/3 and 1/6, 1/3 must be rewritten with a common denominator as 2/6. 7. The reciprocal of 3 is 1/3. 8. 7 divided by sign 4= 7/4

The speed of light is 3×10^8 meters per second, which means that light can travel 300 million meters in just one second. How far can light travel in one minute?

1.8×10^10 meters Correct There are 60 seconds in one minute, so we simply multiply the distance light can travel in one second by 60, which is 6×10^1: 3 × 10^8 meters × 6 × 10^1 =1.8 × 10^10 meters

5^-2=

1/25

One kilometer is equivalent to how many micrometers?

10^9

Recall that 1 foot = 12 inches. Therefore, 1 square foot = _____ square inches.

144

Scientific notation (7×10^4)×(5×10^3)=

3.5 x 10^8

(7×10^2)×(6×10^3)=

4.2×10^6

Thinking about the units of a problem can often help you devise a strategy for solving it. Consider the types of units that go with the following equations, then decide which equations are valid statements.

time=distance/speed distance=speed x time speed=distance/time

10^3×10^−2=

10

Convert a mass of 10^12 micrograms to kilograms.

1000 kilograms

A book written in 1980 states that the peak of the last ice age occurred 18,000 years ago. Therefore, in 2012, the peak had occurred _____ years ago.

18,000 Correct The calculation is 18,000 + 32, since 2012 is 32 years after 1980. The first number (18,000) is precise to the nearest 1000, while the second (32) is precise to the nearest 1. Following the rule for addition and subtraction, the answer should be rounded to the nearest 1000, since that is the precision of the least precise number. Therefore, even though 32 years passed since the book was written, the time since the ice age remains 18,000 years.

When Mars is on the opposite side of the Sun from Earth, it is about 4.0×10^8 km away. How long does it take a radio signal traveling at the speed of light, c=3.00×10^5 km/s, to go from Earth to Mars at this time? Be sure to give your answer with the appropriate precision.

22 minutes

Two days are equivalent to _____ seconds. (Assume exactly 24 hours in one day.)

172,800

A large boulder has a volume of 1000 cubic meters and a mass of 3,200,000 kilograms. What is the density of the boulder, in grams per cubic centimeter?

3.2 g/cm^3

The acceleration of gravity on Mars is about 3.7 meters per second squared. Suppose a rock falls from a tall cliff on Mars. Which of the following equations indicates how fast the rock will be falling after 8 seconds?

3.7 m / s^2 × 8 s

You travel at a speed of about 81 kilometers per hour for about 2.0 hours. Stated with the appropriate precision, you have driven _____ kilometers.

160

Suppose you are asked to find 91,321,972×0.007004. Doing a quick estimate by rounding the numbers in scientific notation, what value would you expect the answer to be close to? (If you need a review of the estimation technique, watch the video Scientific Notation, Part 4.)

63×10^4 Correct Notice that the expression rounds to (9×10^7)×(7×10^−3)= 63×10^7+(−3)=63×10^4

The government in a town of 82,000 people plans to spend $41.5 million this year. Assuming all this money must come from taxes, what is the average number of dollars that the city must collect from each resident?

$510

The mass of the Sun is 1.99×10^30 kilograms and the mass of Earth is 5.97×10^24 kilograms. Therefore, the combined mass of the Sun and Earth is __________.

1.99×10^30 kilograms

China mandates that new cars have an average fuel efficiency of 17.9 kilometers per liter. Given that 1 mile is about 1.6 kilometers, and 1 gallon is about 3.8 liters, choose the equation that gives the equivalent fuel efficiency in miles per gallon.

17.9 km/L x 1 mile/1.6 km x 3.8 L/1 gal

Part B - Unit Conversions Complete the unit conversions by dragging the appropriate labels to their respective targets. Before you begin, you may wish to watch the video Working with Units, Part 2. Drag the appropriate labels to their respective targets. You may use the same label more than once, and not all labels will be used.

2 years x 12 months/1 year = 24 months 24 months x 1 year/12 months = 2 years 6 meters x 100 centimeters/1 meter = 600 centimeters 600 centimeters x 1 meter/100 centimeters = 6 meters

The density of granite is about 2.75 grams per cubic centimeter. Suppose you have a granite countertop for a kitchen that is 1 meter wide, 3 meters long, and 4 centimeters thick. Which of the following equations will give you its mass, in kilograms?

2.75 g/cm^3 x 1 m x 3 m x (100 cm/ 1 m)^2 x4 cm x 1 kg/1000 g

A common measure of health is the body mass index (BMI), defined as a person's weight in kilograms divided by the square of the person's height in meters. That is, BMI= mass (in kg) / [height (in m)]2 Notice that the BMI is a number with no units, but you'll get the correct number only if you use the correct units in the calculation. Consider a man who is 6 feet tall and weighs 200 pounds. What is his body mass index?

27.13

There are 1000 meters in 1 kilometer. The speed of light is 300,000 kilometers per second. Which statement below correctly converts the speed of light to units of meters per second?

300,000 km/s x 1000 m/1 km

Use standard rounding rules to round the number 627.3051 to complete the statements below. You may wish to watch the video Precision and Significant Digits, Part 2. Drag the appropriate labels to their respective targets. You may use the same label more than once, and not all labels will be used.

627.3051 rounded to the nearest 100 is 600 627.3051 rounded to the nearest 10 is 630 627.3051 rounded to 4 significant digits is 627.3 627.3051 rounded to 1 significant digit is 600 627.3051 rounded to the nearest 0.01 is 627.31 627.3051 rounded to the nearest 0.1 is 627.3

The laws of thermal radiation tell us that a star emits light with a total power per unit area that depends only on the star's surface temperature. The formula is power per unit area=σT^4 where σ is a constant with a measured value of σ=5.7×10^−8 W / m^2×K^4 (As usual, m stands for meters and K stands for kelvins.) You can then find the total power (or luminosity) emitted by the star by multiplying its total surface area by the power per unit area. Use these facts to calculate the total power emitted into space by a star with a radius of 8.0×10^8m and a surface temperature of 6500 K. You will need the formula for the surface area of a sphere, which is surface area (sphere)=4πr^2 where r is the radius of the sphere.

8.2×10^26 watts

When you finish a problem, you should confirm that it makes sense and explain it if it does. Suppose you are grading a question in which students are asked to calculate the lifetime of the Sun. A student has submitted an answer of "10^−10 seconds." Which of the following would be the best response to give this student when you grade this question?

Incorrect. Your answer does not make sense, because the time is way too short.

Part A - Using Metric Units Complete each statement using the appropriate metric unit. Before you begin, you may wish to watch the video Metric Units, Part 1. Drag the appropriate labels to their respective targets. You may use the same label more than once, and not all labels will be used.

My friend is 2 METERS tall. My water bottle has a volume of 1.5 LITERS. My thumb is 2 CENTIMETERS wide. I bought 3 KILOGRAMS of apples at the store. A penny weighs about 3 GRAMS. I can run 60 METERS in 10 seconds.

Part B - Definition of Roots Based on the definitions of roots, use the words on the left to complete the sentences on the right. The video Powers and Roots, Part 2, reviews the ideas you'll need for this part. Drag the appropriate labels to their respective targets. You may use the same label more than once, and not all labels will be used.

Square root of 25 asks what number to the SECOND power makes 25. 25^1/2 asks what number to the SECOND power makes 25. 3 square root 8 asks what number to the THIRD power makes 8. 8^1/3 asks what number to the THIRD power makes 8. The square root of 16 asks what number to the SECOND power makes 16. 4 square root 16 asks what number to the FOURTH power makes 16. Square root of 1/4 asks what number to the SECOND power makes 1/4.

Part C - Comparing Powers of 10 Powers of 10 make it very easy to write large and small numbers, but as a result it can also be easy to forget the large differences between different powers. We can compare any two numbers by dividing them. For example, we say that 12 is four times as large as 3 because 12÷3=4. Complete the sentences below comparing pairs of powers of 10. Drag the appropriate labels to their respective targets. Note: You may use the same label more than once, and not all labels will be used.

10^80 is 10 times as large as 10^79 10^-1 is 1000 times as large as 10^-4 10^-2 is 1/10 as large as 10^-1 10^-4 is 1/1000 as large as 10^-1 10^5 is 100 times as large as 10^3 10^2 is 10,000 times as large as 10^-2 10^10 is 1,000,000,000 times as large as 10^1

A scale in the doctor's office that can be read to the nearest kilogram shows your mass as 63 kilograms. You then write down that your mass is "63.00 kilograms." Your statement is__________.

misleading, because it implies that you know your mass to greater precision than you actually do

Part A - Understanding Units Complete the statements by dragging the appropriate labels to their respective targets. Before you begin, you may wish to watch the video Working with Units, Part 1. Drag the appropriate labels to their respective targets. You may use the same label more than once, and not all labels will be used.

In words, m/s means METERS PER SECOND. In words, m^2 means SQUARE METERS. In words, m^3 means CUBIC METERS. In words, kg/m^3 means KILOGRAMS PER CUBIC METER. In words, m/s^2 means METERS PER SECOND SQUARED. In words, 4 boxes x 5 kg/box means 4 BOXES OF 5 KILOGRAMS PER BOX.