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Figure 1-1e Figure 1-1f Domain: 0 speed speed limit Range: time minimum time at speed limit This problem set will help you see the relationship between variables in the real
world and functions in the mathematical world. Problem Set 1-1
1. Archery Problem 1: An archer climbs a tree nearthe edge of a cliff, then shoots an arrow high
into the air. The arrow goes up, then comes
back down, going over the cliff and landing in
the valley, 30 m below the top of the cliff. The
arrow’s height, y, in meters above the top of
the cliff depends on the time, x, in seconds,
since the archer released it. Figure 1-1g shows
the height as a function of time. Figure 1-1g a. What was the approximate height of the arrow at 1 second? At 5 seconds? How do you explain the fact that the height is
negative at 5 seconds?
b. At what two times was the arrow at 10 m above the ground? At what time does the
arrow land in the valley below the cliff?
c. How high was the archer above the ground at the top of the cliff when she released the
arrow?
d. Why can you say that altitude is a function of time? Why is time not a function of
altitude?
e. What is the domain of the function? What is the corresponding range?
2. Gas Temperature and Volume Problem: When you heat a fixed amount of gas, it expands,
increasing its volume. In the late 1700s, French
chemist Jacques Charles used numerical
measurements of the temperature and volume
of a gas to find a quantitative relationship
between these two variables. Suppose that
these temperatures and volumes had been
recorded for a fixed amount of oxygen.
Pietro Longhi’s painting, The
Alchemists, depicts a laboratory setting
from the middle of the 18th century. a. On graph paper, plot V
as a function of T.
Choose scales that go at
least from T = –300 to
T = 400. b. You should find, as
Charles did, that the
points lie in a straight
line! Extend the line
backward until it crosses the T-axis. The
temperature you get is called absolute zero,
the temperature at which, supposedly, all
molecular motion stops. Based on your
graph, what temperature in degrees Celsius
is absolute zero? Is this the number you
recall from science courses?
c. Extending a graph beyond all given data,
as you did in 2b, is called extrapolation.
“Extra-” means “beyond,” and “-pol-” comes
from “pole,” or end. Extrapolate the graph
to T = 400 and predict what the volume
would be at 400°C.
d. Predict the volume at T = 30°C. Why do you
suppose this prediction is an example of
interpolation?
e. At what temperature would the volume be
5 liters? Which do you use, interpolation or
extrapolation, to find this temperature?
e. At what temperature would the volume be
5 liters? Which do you use, interpolation or
extrapolation, to find this temperature?
f. Why can you say that the volume is a
function of temperature? Is it also true that 0 9.5 50 11.2 100 12.9 150 14.7 200 16.4 250 18.1 300 19.9
Extrapolation (p. 6): Using a function to estimate a value outside the range of the given data.
© 2003 Key Curriculum Press. The individual purchaser of this HTML edition of Precalculus with Trigonometry:
Concepts and Applications may print single copies of pages for individual use. All other forms of reproduction,
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T (°C) V (l)
© 2003 Key Curriculum Press
Interpolation (p. 6): Using a function to estimate a value within the range of given data.
the temperature is a function of volume?
Explain.
g. Considering volume to be a function of
temperature, write the domain and the
range for this function.
h. See if you can write an algebraic equation
for V as a function of T.
i. In this problem, the temperature is the
independent variable and the volume is the
dependent variable. This implies that you can
change the volume by changing the
temperature. Is it possible to change the
temperature by changing the volume, such
as you would do by pressing down on the
handle of a tire pump?
3. Mortgage Payment Problem: People who buy
houses usually get a loan to pay for most of the house and pay on the resulting mortgage each month. Suppose you get a $50,000 loan and pay it back at $550.34 per month with an interest rate of 12% per year (1% per month). Your balance, B
dollars, after n monthly payments is given by the algebraic equation
a. Make a table of your balances at the end of
each 12 months for the first 10 years of the
mortgage. To save time, use the table feature
of your grapher to do this.
b. How many months will it take you to pay off
the entire mortgage? Show how you get your
answer.
rofl rofl
 lov2catnap · Fri Aug 17, 2007 @ 02:18am · 0 Comments |
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