1–6 Convert the following into the new units.
1. Find 3.4 pounds in grams (1 ounce is 28.35 grams and 1 pound is 16 ounces).
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2. Find 1 yard in mm (1 in. is 25.40 mm and 1 yard is 36 in.).
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3. Find 60 years in hours (1.0 year ˜ 365.25 days).
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4. Find 65 miles per hour in centimeters per second.
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5. Find 2.3 grams per cubic centimeter in pounds per cubic foot.
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6. Find 9.807 m/sec\cf0 cf2 2 (the acceleration of gravity) in miles per hour per second.
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7-10 Compute the answers by adding the given quantities.
7. A boy who is 1.34 meters tall grows 2.3 cm.
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8.
After waiting for 1.2 hours for a plane flight, you are told you will
have to wait another 17 minutes. What is the total wait?
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9.
You purchase 6 apples that weigh 145 g each, and 7 oranges that weigh
123 g each. What is the total weight if you add the apples to the
oranges?
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10.
The density of the apples in the previous problem is 0.8 g/cm 3 and the
density of the oranges is 0.95 g/cm 3 . What is the total volume if you
add the apples to the oranges?
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11-14 Figure out which of the following is larger.
11. The area of a square with side length 1.7 cm or of a disk with radius 1.0 cm.
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12. The perimeter of a square with side length 1.7 cm or of a circle with radius 1.0 cm.
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13. The volume of a sphere with radius 100 m or of a 50 cm deep lake with an area of 3.0 square km.
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14. The surface area of a sphere with radius 100 m or the surface area of a lake with area 3.0 square km.
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15-18 Find the dimensions of the following quantities.
15. Pressure (force per unit area)
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16. Energy (force times distance)
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17. The rate of change of the area of a colony of bacteria growing on a plate.
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18.
The force of gravity between two objects is equal to Gm 1 m 2 /r 2
where m 1 and m 2 are the masses of the two objects, and r is the
distance between them. What are the dimensions of the gravitational
constant G?
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19-22 Check whether the following formulas are dimensionally
consistent.
19. Distance = rate times time.
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20. Velocity = acceleration times time.
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21. Force = mass times acceleration.
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22.
Energy = 1/2 mass times the square of velocity (see Exercise 16 for the
units of energy).
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23-26 Using the graph of the function g ( x ), sketch a graph of the
shifted or scaled function, say which kind of shift or scale it is, and
compare with the original function.
23. 4g ( x )
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24. g ( x )- 1
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25. g ( x /3)
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26.
g ( x + 1)
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27-30 Find the volumes of the following two cartoon trees (drawing a
sketch can help) assuming that the height of the first is 23.1 m and
that the height of the second is 24.1 m. What is the ratio of the volume
of the larger tree to that of the smaller tree?
27.
A tree is a perfect cylinder with radius 0.5 m no matter what the
height (the volume of a cylinder with height h and radius r is p hr 2 ).
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28. A tree is a perfect cylinder with radius equal to 0.1 times the height.
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29.
A tree looks like the tree in Exercise 27 , but with half the height in
the cylindrical trunk and the other half in a spherical blob on top.
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30.
A tree looks like the tree in Exercise 27 , but with 90% of the height
in the cylindrical trunk and the remaining 10% in a spherical blob on
top.
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31-34 Find the mass in kilograms of the following objects.
31. A water bed that is 2 m long, 20 cm thick, and 1.5 m wide. The density of water is 1.0 g/cm 3 .
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32. A spherical cow with diameter 1.3 m and density 1.3 g/cm 3 .
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33. A coral colony of 3200 individuals each weighing 0.45 g.
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34. A circular colony of mold with diameter of 4.8 cm and density of 0.0023 g/cm 2 .
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35-38 Change the units in the following functions, and compare a graph in the new units with the original units.
35.
(Based on Section 1.2, Exercise 45 ) The number of bees b on a plant is
given by b = 2 f + 1 where f is the number of flowers. Suppose each
flower has 4 petals. Graph the number of bees as a function of the
number of petals.
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36.
The number of cancerous cells c as a function of radiation dose r is c =
r - 4 for r (measured in rads) greater than or equal to 5, and is zero
for r less than 5 (as in Section 1.2, Exercise 46 ). Suppose that
radiation is instead measured in millirads (1 rad = 1000 millirads).
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37.
Insect development time A (in days) obeys A = 40 - T/ 2 where T
represents temperature in °C for °C between 10 and 40 (as in Section
1.2, Exercise 47 ). Suppose that development time is measured in hours.
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38.
Tree height h (in meters) follows the formula where a represents the
age of the tree in years (as in Section 1.2, Exercise 48 ). Suppose that
tree age is measured instead in decades.
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39-44 Estimate the following.
39.
The speed of light in cm per ns (10 -9 seconds or one nanosecond) (the
speed of light is about 186,000 miles/second). A fast computer takes
about 0.3 ns per operation. How far does light travel in the time
required by one operation?
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40. Estimate the speed that your hair grows in miles per hour. (This problem was borrowed from the book Innumeracy .)
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41.
The weight of the earth in kilograms. The earth is approximately a
sphere with radius 6500 km and density 5 times that of water.
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42.
Suppose a person eats 2000 Kcal per day. Using the facts that 1 Kcal is
approximately 4.2 Kj (a kilojoule is a unit of energy equal to 1000
joules) and 1 watt is one joule per second (a unit of power), about how
many watts does a person use?
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43. If a movie is about 2 hours long, how many movies could you watch if you spent half your time watching movies for 60 years?
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44.
The volume of all the people on earth in cubic kilometers. If a large
mine is about 3 km across and 1 km deep, would they all fit?
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45-48 The following problems give several ways to estimate the number of
cells in your body. A cell is roughly a sphere 10 µ m in radius, where 1
µ m is 10 -6 m.
45.
Using the fact that the density of a cell is approximately the density
of water and that water weighs 1 g/cm 3 , estimate the number of cells
in your body.
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46.
Estimate your volume in cubic meters by pretending you are shaped like a
board. Pretending that cells are cubes 20 µ m on a side, what do you
estimate the number of cells to be by this method?
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47.
The brain weighs about 1.3 kg and is estimated to have about 100
billion neurons and 10 to 50 times as many other cells (glial cells). Is
this consistent with our previous estimates?
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48.
The nematode C . elegans is a cylinder about 1 mm long and 0.1 mm in
diameter, consisting of about 1000 cells. Are these cells about the same
size as the ones in your body?
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49–50 The following problems regard tying string or gift-wrapping our
planet, thought of as a sphere with radius 6500 km.
49.
How long a piece of string would be required to go around the equator?
If the string were made 1.0 meters longer and stretched out all the way
around, how high would it be above the surface? Does the result surprise
you?
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50.
How large a piece of shrink-wrap would be required to cover the entire
planet? If the wrap were increased in area by 1.0 m 2 and stretched out
all around, how high would it be above the surface? Why do you think the
result is so different from the previous problem? (Working this out
takes a lot of decimal places.)
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