4.2 Exercises
1) Postage for a package is $3 for the first pound and $2 for each additional pound.
a) Model the postage price, y , as a function of weight, x, in pounds using a linear equation.
b) What is the slope and what does it tell us about postage price? Be specific and include units in your explanation.
c) What is the y-intercept and what does it tell us about postage price? Be specific and include units in your explanation.
2) The relationship between degrees Fahrenheit (F) and degrees Celsius (C) is linear. Water freezes at 0°C and 32°F. Water boils at 100°C and 212°F.
a) Write an equation that expresses Fahrenheit temperature F as a linear function of Celsius temperature C.
b) Determine the slope and explain what it tell us about degrees.
3) Kisha’s camper has a 20 gallon gas tank and she gets 12 miles to the gallon (that is, she uses 1/12 gallon per mile).
a) Complete the table of values for the amount of gas, g, left in Kisha’s tank after driving m miles, assuming she starts with a full tank.
| m | 0 | 48 | 96 | 144 | 192 |
|---|---|---|---|---|---|
| g |
b) Write an equation that expresses the amount of gas, g, left in Kisha’s tank in terms of the number of miles, m, she has driven.
c) Graph the equation below
d) How much gas will Kisha use between when her odometer reads 96 miles and when the odometer reads 144 miles? Illustrate this on the graph.
e) If Kisha has less than 5 gallons of gas left, how many miles has she driven? Illustrate this on the graph.
4) Mayuri works as a waiter in a restaurant. He earns $1,400 per month as a base salary, plus tips averaging 15% of the total cost of the meals he served that month.
a) Let S represent how much Mayuri earns in one month and C represent the total monthly meal cost. Write a linear equation expressing S in terms of C.
b) Determine the slope of the function and explain what this number tells us about Mayuri’s monthly earnings.
c) Determine the y-intercept of the equation and explain what this point tells us about Mayuri’s monthly earnings.
5) Consider the linear equation [latex]2𝑥 + 5𝑦 = 30[/latex]. Write a story problem that would result in this equation (something that gives meaning to the numbers 2, 5 and 30). Then, state what the variables 𝑥 and 𝑦 represent in terms of your story problem scenario.
6) There is a population of 200 tigers in a national park. Unfortunately, they are being illegally poached at the rate of 8 tigers per year. Assume the population is otherwise unchanging:
a) Write a linear model for the tiger population, P, in t years.
b) Graph your linear model from part (a).
c) Determine the t-intercept and explain what it tells us about the tiger population.
d) Determine the P-intercept and explain what it tells us about the tiger population.
e) Find how many years would it take for these tigers to become extinct in this park?
f) How many years would it take to reduce the number of tigers in this park to half of the current population?
The slope of a function from one input value to another input value is the change in corresponding output values divided by the change in the given input values (often known as the change in y divided by the change in x or "rise over run"). Graphically, the slope gives the steepness of the line through the two corresponding points.
The y-intercept of function is the point at which the graph of the function intersects the y-axis (where x = 0).
A function whose graph is a straight line (constant slope). Such functions may be written in the form y = mx+b where m is the slope and (0,b) is the y-intercept.
