Download Black - Divide by Whole Numbers and Powers of 10 Scientific Notation

Black - Divide by Whole Numbers and Powers of 10
Scientific Notation (continued from last lesson)
Order from least to greatest.
1. 8 x 10 -­‐ 9, 14.7 x 10 -­‐ 7, 0.22 x 10 -­‐ 10
You can multiply numbers in scientific notation using the rule for Multiplying Powers with
the Same Base.
Example
Multiply 3 x 10 -­‐
7
and 9 x 10 3. Express the result in scientific notation.
(3 x 10 -­‐ 7)(9 x 10 3) = 3 x 9 x 10 -­‐ 7 x 10 3 Use the Commutative Property of Multiplication.
= 27 x 10 -­‐ 7 x 10 3
Multiply 3 and 9.
4
= 27 x 10 -­‐
Add the exponents.
1
4
= 2.7 x 10 x 10 -­‐
Write 27 as 2.7 x 10 1
= 2.7 x 10 -­‐ 3
Add the exponents.
Multiply. Express the result in scientific notation.
2. (7.1 x 10 -­‐ 8)(8 x 10 4)
3. Chemistry. A hydrogen atom has a mass of 1.67 x 10 -­‐ 27 kg.
What is the mass of 6 x 10 3 hydrogen atoms? Express the result in scientific
notation.
Use a calculator to enter these numbers in standard form, and work out the
problems. Leave the answer in Scientific Notation.
4. (2.9 x 10 -­‐
) x (7.8 x 10 8 )
13
5. (1.532 x 10 4) x (9.883 x 10 -­‐ 5)
6. (6.982 x 10 8)/(3.118 x 10 -­‐ 4)
7. (4.86 x 10 -­‐ 3)/(2.04 x 10 8 )
8. (8.7552 x 10 2)/(9.318 x 10 -­‐
11
)
More Problems
9. Scientific Sums. Mathematicians and scientists often use scientific notation when
working with very large or very small numbers. The number 25,700,000,000,000,000
can be written in scientific notation as 2.57 x 1016. Now isn't that a space saver?
When it comes to adding numbers written in scientific notation, you must be very
careful with place value. Please help me to find the sum of the following numbers:
2.57 x 1016
3.64 x 1015
7.93 x 1016
4.83 x 1015
Don't forget to write your final answer in a complete sentence. Also, be sure that
your sum is written in scientific notation.
Bonus: When working with very large or very small numbers, mathematicians and
scientists will also consider significant digits.* Explain what is meant by significant
digits and state your answer to the sum using significant digits.
10. The speed of light is 186,000 miles per second. How many miles per hour is the speed
of light? Express your answer in scientific notation to one decimal place.
Problem Involving Really Big or Small Numbers
11. Clothing Industry Problem. There are about 230 million people in the United States.
Assume that each person buys 6 pairs of socks a year and that socks cost about $4 a
pair. How much money can the clothing industry expect to take in each year from
selling socks?
12. Automotive Batteries Problem. There are about 120 million cars and trucks in the
United States. Each one uses a battery, which needs replacing about once every 3
years. If an average battery costs $80, how much money can the battery
manufacturing industry expect to make each year?
13. Glacier Problem. A typical glacier moves downhill at about 5 inches per day. Suppose
that the glacier is 4 miles long.
a. Recalling that 1 mile is 5280 feet, how many days will it take for snow that falls
at the top of the glacier to move downhill to the bottom?
b. How many years in this?
14. Numbers on a Calculator Problem. A normal calculator can handle numbers as big as
9.99 x 1099. To see how big a number this is, answer the following questions.
a. The volume of the earth is 2.59 x 1011 cubic miles. One cubic mile is 52803 cubic
feet. What is the volume of the earth in cubic feet?
-9
b. One grain of sand has a volume of about 1.3 x 10 cubic feet. Divide the answer
in part (a) by 1.3 x 10-9 to find the number of grains of sand the earth could hold.
c. Which is bigger, the number of grains of sand the earth could hold or the largest
number a calculator can hold?
15. Sahara Desert Problem. The Sahara Desert covers about 3 million square miles.
a. Write this number in scientific notation.
b. One square mile is 52802 square feet. About how many square feet in the Sahara
Desert?
c. Assume that the Sahara Desert is covered with sand to an average depth of 200
feet. The volume of this sand in cubic feet is the area in square feet times the
depth in feet. How many cubic feet of sand are there?
d. A grain of sand has a volume of about 1.3 x 10-9 cubic foot. Divide the volume in
cubic feet by 1.3 x 10-9 to find out approximately how many grains of sand there
are in the Sahara Desert.
16. Sizing Up Sequoias. A sequoia tree seed weighs only 1/(5 x 103) of an ounce.
If a mature sequoia tree weighs an average of 2.16 x 1011 times
as much, how much does the average mature sequoia weigh?
Give the weight in pounds; there are 16 ounces in a pound.
Extra: If I weigh 100 lbs, how many seeds would it take to
equal my weight? How many of me would it take to equal the
weight of the tree?
17. If a circle has a radius of 4.5 inches, what is the length of 1° of its circumference?
(Round to the nearest hundredth.)
18. A square piece of paper has a perimeter of 38.25 after it is folded in half. What was
the length of each side of the square before it was folded?
19. An Isosceles triangle with a height of 9.375 inches is folded along its only line of
symmetry. The perimeter of the folded triangle is 23.75 inches. What is the
perimeter of the original, triangle?
Solutions
1. 2.2 x10−11 ,8 x10−9 ,1.47 x10−6
2. 7.1x8 x10−8 x104 = 56.8 x10−4 = 5.68 x10−3
3. 1.67 x6 x10−27 x103 = 10.02 x10−24 = 1.002 x10−23
4. 2.9 x7.8 x10−13 x108 = 22.62 x10−5 = 2.262 x10−4
5. 1.532 x9.883x104 x10−5 = 15.14 x10−1 = 1.514 x100
6. 6.982 x3.118 x108 x10−4 = 21.77 x104 = 2.177 x105
7. 4.86 x 2.04 x10−3 x108 = 9.9144 x105
8. 8.7552 x9.318 x102 x10−11 = 81.58 x10−9 = 8.518 x10−8
9. Scientific Sums. The sum of the four numbers in scientific notation is 1.1347 X 1017.
Bonus: The sum using significant digits is 1.134 X 1017.
We changed all the numbers in scientific notation to 1015. Then we added them.
25.7
3.64
79.3
+ 4.83
-----113.47
X 1015
X 1015
X 1015
X 1015
X 1015
Since in scientific notation you can only have one number in front of the decimal, we
moved the decimal two places to the left. We also added two exponents because we
moved the decimal two places. Our final answer was 1.1347 X 1017.
Bonus: Significant digits are digits you can trust to be accurate.
We redid the original problem. We put a variable for the decimal places that we
didn't know.
25.7xx
3.64x
79.3xx
+ 4.83x
-----113.4xx
X 1015
X 1015
X 1015
X 1015
X 1015
Because there are X's in the hundredths place, we don't know the result of the
hundredths place, so we dropped the seven. We once again moved the decimal two
places to the left and added two exponents. Our answer in significant digits is 1.134
X 1017.
10. If light travels 186,000 miles in 1 second, then it travels 186,000 x 3600 =
669,600,000 or 6.7 x 108 miles per hour.
11. 5,520,000,000
12. About 3.2 billion dollars!
13. a. 50688 days
b. 139 years
14. a. 1.36752 x 1015
b. 1.051938462 x 1024
c. calculator
15. a.
b.
c.
d.
3 x 106
8 x 1013 sq ft
1.7 x 1016 cu ft
1.3 x 1025 grains
16. Sizing Up Sequoias. First of all, I figured out that 1/(5 x 103) was 1/5000 of an
ounce.
If one seed only weighs 1/5000 of an ounce then 5000 seeds would make an ounce.
To figure out what an average sequoia tree weighs I did:
1/5000x(2.16 x 1011)
43,200,000 oz.
To convert that to pounds I did: 43,200,000 divided by 16 because there are
sixteen oz. in a pound. That equals 2,700,000.
To convert that to tons I did: 2,700,000 divided by 2000 because there are 2000
lbs in a ton. That equals 1,350.
Therefore, the average mature sequoia weighs 1,350 tons.
EXTRA:
To find how many seeds it would take to equal 100 pounds I did
5000(seeds to an ounce)x16(to make a pound) x 100 (to make 100 pounds)
5,000x16x100
80,000x100
8,000,000
It would take 8,000,000 seeds to equal 100 pounds.
Since the tree weighs 2,700,000 pounds and you weigh 100 pounds, I found the
answer by dividing 2,700,000 by 100.
2,700,000
--------- = w
100
27,000=w
It would take 27,000 of you to equal the weight of a sequoia tree.
17. .08 inches
A circle with a 4.5 inch radius has a 9 inch diameter. The formula for the
circumference of a circle is pi x diameter: 9 x 3.14 = 28.26 inch circumference.
Because there are 360 degrees in a circle, the length of one degree is
28.26 inches ÷ 360 = .0785 inches
18. When the paper is folded in half, its short sides are equal to half of the original side
of the square. Add up the sides of the folded rectangle: 3n = 38.25 n = 12.75 inches
19. 28.75 inches
When the triangle is folded in half, its perimeter is 23.75 inches. Because the height
is known, the remaining two sides of the triangle must equal 23.75 - 9.375 =14.375
inches.
The original triangle was twice this length: 2 x 14.375 = 28.75 inches.
Bibliography Information
Teachers attempted to cite the sources for the problems included in this problem set. In some cases,
sources may not have been known.
Problems
Bibliography Information
9, 16
The Math Forum @ Drexel
(http://mathforum.org/)
17-19
Zaccaro, Edward. Challenge Math (Second
Edition): Hickory Grove Press, 2005.