College Algebra 9th txtbk

150 CHAPTER 2 GRAPHS
EXAMPLE 3
Mixing Antifreeze
Ethylene glycol and propylene glycol are liquids used in antifreeze and deicing solutions.
Ethylene glycol is listed as a hazardous chemical by the Environmental Protection Agency,
while propylene glycol is generally regarded as safe. Table 1 lists solution concentration
percentages and the corresponding freezing points for each chemical.
Table 1
Concentration Ethylene Glycol Propylene Glycol
20% 15°F 17°F
50% 36°F 28°F
(A) Assume that the concentration and the freezing point for ethylene glycol are linearly
related. Construct a linear model for the freezing point.
(B) Interpret the slope in part (A).
(C) What percentage (to one decimal place) of ethylene glycol will result in a freezing
point of 10°F?
SOLUTIONS
(A) We begin by defining appropriate variables:
Let
p percentage of ethylene glycol in the antifreeze solution
f freezing point of the antifreeze solution
From Table 1, we see that (20, 15) and (50, 36) are two points on the line relating
p and f. The slope of this line is
and its equation is
m f 2 f 1 15 (36)
51
p 2 p 1 20 50 30 1.7
f 15 1.7( p 20)
f 1.7p 49
Linear model
(B) The rate of change of the freezing point with respect to the percentage of ethylene
glycol in the antifreeze solution is 1.7 degrees per percentage of ethylene glycol.
Increasing the amount of ethylene glycol by 1% will lower the freezing point by 1.7°F.
(C) We must find p when f is 10°.
f 1.7p 49
10 1.7p 49
1.7p 59
p 59
1.7 34.7%
Add 10 1.7p to both sides.
Divide both sides by 1.7.
MATCHED PROBLEM 3 Refer to Table 1.
(A) Assume that the concentration and the freezing point for propylene glycol are linearly
related. Construct a linear model for the freezing point.
(B) Interpret the slope in part (A).
(C) What percentage (to one decimal place) of propylene glycol will result in a freezing
point of 15°F?