College Algebra 9th txtbk

SECTION 2–4 Linear Equations and Models 155
2-4 Exercises
1. Explain the steps that are involved in the process of mathematical
modeling.
2. If two variables x and y are linearly related, explain how to calculate
the rate of change.
3. If two variables x and y are not linearly related, explain how to
calculate the average rate of change from x x 1 to x x 2 .
4. Explain the difference between interpolation and extrapolation
in the context of regression analysis.
APPLICATIONS
5. COST ANALYSIS A plant can manufacture 80 golf clubs per day
for a total daily cost of $8,147 and 100 golf clubs per day for a
total daily cost of $9,647.
(A) Assuming that daily cost and production are linearly related,
find the total daily cost of producing x golf clubs.
(B) Interpret the slope of this cost equation.
(C) What is the effect of a 1 unit increase in production?
6. COST ANALYSIS A plant can manufacture 50 tennis rackets per
day for a total daily cost of $4,174 and 60 tennis rackets per day for
a total daily cost of $4,634.
(A) Assuming that daily cost and production are linearly related,
find the total daily cost of producing x tennis rackets.
(B) Interpret the slope of this cost equation.
(C) What is the effect of a 1 unit increase in production?
7. FORESTRY Forest rangers estimate the height of a tree by measuring
the tree’s diameter at breast height (DBH) and then using a
model constructed for a particular species.* A model for white
spruce trees is
h 4.06d 24.1
where d is the DBH in inches and h is the tree height in feet.
(A) Interpret the slope of this model.
(B) What is the effect of a 1-inch increase in DBH?
(C) How tall is a white spruce with a DBH of 12 inches? Round
answer to the nearest foot.
(D) What is the DBH of a white spruce that is 100 feet tall? Round
answer to the nearest inch.
8. FORESTRY A model for black spruce trees is
h 2.27d 33.1
where d is the DBH in inches and h is the tree height in feet.
(A) Interpret the slope of this model.
(B) What is the effect of a 1-inch increase in DBH?
(C) How tall is a black spruce with a DBH of 12 inches? Round answer
to the nearest foot.
(D) What is the DBH of a black spruce that is 100 feet tall? Round
answer to the nearest inch.
*Models in Problems 7 and 8 are based on data found at
http://flash.lakeheadu.ca/~fluckai/htdbh04.xls
9. Dr. J. D. Robinson and Dr. D. R. Miller published the following
models for estimating the weight of a woman:
Robinson: w 108 3.7h
Miller: w 117 3.0h
where w is weight (in pounds) and h is height over 5 feet (in inches).
(A) Interpret the slope of each model.
(B) If a woman is 56 tall, what does each model predict her weight
to be?
(C) If a woman weighs 140 pounds, what does each model predict
her height to be?
10. Dr. J. D. Robinson and Dr. D. R. Miller also published the following
models for estimating the weight of a man:
Robinson: w 115 4.2h
Miller: w 124 3.1h
where w is weight (in pounds) and h is height over 5 feet (in inches).
(A) Interpret the slope of each model.
(B) If a man is 510 tall, what does each model predict his weight
to be?
(C) If a man weighs 160 pounds, what does each model predict his
height to be?
11. SPEED OF SOUND The speed of sound through the air near sea
level is linearly related to the temperature of the air. If sound travels
at 741 mph at 32°F and at 771 mph at 72°F, construct a linear
model relating the speed of sound (s) and the air temperature (t). Interpret
the slope of this model.
12. SPEED OF SOUND The speed of sound through the air near sea
level is linearly related to the temperature of the air. If sound
travels at 337 mps (meters per second) at 10°C and at 343 mps at
20°C, construct a linear model relating the speed of sound (s)
and the air temperature (t). Interpret the slope of this model.
13. SMOKING STATISTICS The percentage of male cigarette
smokers in the United States declined from 25.7% in 2000 to
23.9% in 2006. Find a linear model relating the percentage m of
male smokers to years t since 2000. Use the model to predict the
first year for which the percentage of male smokers will be less
than or equal to 18%.
14. SMOKING STATISTICS The percentage of female cigarette
smokers in the United States declined from 21.0% in 2000 to
18.0% in 2006. Find a linear model relating the percentage f of
female smokers to years t since 2000. Use the model to predict the
first year for which the percentage of female smokers will be less
than or equal to 10%.
15. BUSINESS—DEPRECIATION A farmer buys a new tractor for
$142,000 and assumes that it will have a trade-in value of $67,000
after 10 years. The farmer uses a constant rate of depreciation (commonly
called straight-line depreciation—one of several methods
permitted by the IRS) to determine the annual value of the tractor.
(A) Find a linear model for the depreciated value V of the tractor t
years after it was purchased.