Sections 9A,9B
Functions, Linear Functions.
III (Examples)
Does it make sense?
I graphed a function showing how my heart rate depends on my running speed. The domain was heart rates from 60 to 180 beats per minute.
Solution. The answer is no, it does not make sense. In this case, the independent variable is the running speed, so the domain should include the relevant running speed values, not the heard rate values.
Does it make sense?
It is possible to make a linear model from any two data points, but there's no guarantee that the model will fit other data points.
Solution. The answer is yes. Indeed, we know that given any two distinct points, we can draw a unique straight line through them. If we have more than two data points, they clearly may not lie on the same line.
Exercise 14, p524
Suppose you walk through a used car lot and list the price and model year of each car you see. Are the corresponding two variables related in a way that can be described as a function? If so, identify the dependent and independent variables.
Solution. Clearly, the same price may correspond to several model years, so the model year cannot be a function of the price. Also, several cars made in the same year can have different prices, so the price cannot be a function of the model year either.
Consider the graph.
a. In words, describe the function shown on the graph.
Answer. The record pace decreases linearly with the length of race.
b. Find the slope of the graph, and express it as a rate of change,
Answer. The slope is approximately −2010=−2. The rate of change is -2 (km/hr)/km.
A gas station owner finds that for every penny increase in the price of gasoline, she sells 80 gallons fewer of gas per week. How much more or less gas will she sell if she raises the price by 8 cents per gallon?
Solution.
At this gas station, the amount of gas sold depends linearly on the price. The rate of change is
−80
(gallons per week)/cent.
If the price increases by 8 cents, the change is
−80×8=−640
That is a 640 gallons less sold per week.
You run along a path at a constant speed of 5.5 miles per hour. How far do you travel in 1.5 hours?
Solution.
The distance traveled is a linear function of time.
It increases at
a constant rate of 5.5
miles per hour.
The distance traveled in 1.5 hours is
5.5×1.5=8.25
miles.
The cost of leasing a car is $1,000 for a downpayment and processing fee plus $360 per month. For how many months can you lease a car with $3680?
Solution. The total amount spent y is a linear function of time x with a rate of change of 360, and initial value (y-intercept) of 1000. This function can be written as
y=360×x+1000
Solve the equation for x: x=y−1000360, and find for y=3680:
x=3680−1000360≈7.44. That means 7 full months of lease.
Recall that linear function is described by the formula
y=mx+b
This equation can always be solved for x:
x=y−bm
and we always can find the value of independent variable x
for a given value of the dependent variable y
The cost of leasing a car is $1,000 for a downpayment and processing fee plus $360 per month. For how many months can you lease a car with $3680?
Alternative solution.
After paying $1,000 of downpayment and processing fee you are left with
3680−1000=$2680.
With a monthly payment of $360, that suffices for
2680360≈7.44,
or 7 full months.
You can purchase a motorcycle for $6,500 or lease it for a downpayment of $200 and $150 per month. Find the function which describes how the cost of the lease depends on time. How long can you lease the motorcycle before you've paid more than its purchase price?
Solution. This is a linear function with a slope of 150, and an initial value (y-intercept) of 200.
The equation for this function thus is
y=150x+200.
You can purchase a motorcycle for $6,500 or lease it for a downpayment of $200 and $150 per month. Find the function which describes how the cost of the lease depends on time. How long can you lease the motorcycle before you've paid more than its purchase price?
Solution. We found the linear function y=150x+200.
Solve the equation for x:
x=y−200150,
and find for y=6500
x=6500−200150=42 months (3.5 years) of lease.
You can purchase a motorcycle for $6,500 or lease it for a downpayment of $200 and $150 per month. How long can you lease the motorcycle before you've paid more than its purchase price?
Alternative solution (no function involved).
After $200 of downpayment is paid, you are left with
6500−200=$6300.
With a monthly lease of $150, this amount suffices for
6300150=42 months (equivalently, 3.5 uears) of lease.
A mining company can extract 2000 tons of gold ore per day with a purity of 3 ounces of gold per ton. The cost of extraction is $1000 per ton. If p is the price of gold in dollars per ounce, find the function that gives the daily profit/loss of the mine as it varies with the price of gold.
Solution. While the total cost of extraction is 1000×2000=2,000,000 dollars, the price of all gold extracted during a day is 2000×3×p=6000×p dollars.
The daily profit P is the difference between them:
P=6000p−2000000
What is the minimum price of the gold which makes the mine profitable?
Solution. We found the profit P as a function of the price p:
P=6000p−1200000
This becomes a loss, and the mine stops to be profitable when P=0:
0=6000p−2000000.
Solve for p to find p=20000006000≈$333.33
Exercise 346, p538
The amount of sugar in a fermenting batch of beer decreases with time at the rate of 0.1 gram per day, starting from an initial amount of 5 grams. When is the sugar gone?
Solution.
Let y denote the amount of sugar in the batch. It is a
linear function of time, t. We know that the initial amount
is 5 grams and the rate of change is −0.1 gram per day. So, the
formula is:
y=5−0.1t
We need to find the value of time when y is zero, so
0=5−0.1t, giving us t=50.1=50 days.