Number, Relations and Functions 10
A
PPLICATIONS
1. The Ace Plumbing Company advertises that it will take approximately 4 hours for its 180 L home hot water tank to heat cold water to the required hot water temperature. An inspector filled the tank with cold water and found that after 2 hours the temperature of the water was 35 degrees Celsius and after 3 hours, 47.5 degrees Celsius.
(a) Plot a graph of this relation.
(b) Find the equation for this relation expressing temperature in terms of time.
(c) What quantity does the slope of the relation represent?
(d) What will be the water temperature after 1 hour of heating?
(e) Determine the y-intercept. What meaning does the intercept have?
(f) Determine how long it will take for the water to reach 50 degrees Celsius.”
(g) Why is the graph of the relation limited to the first quadrant?
(h) State the domain and range of the relation.
(i) Why would a maximum limit be put on the water temperature?
2. The student council decides to provide a band and a lunch for the graduation dance. If 200 people attend, the cost is $1600. If 300 people attend, the price increases to $1900.
(a) Plot a graph of this relation.
(b) Find an equation of this relation expressing cost
in terms of the number of people attending.
(c) The fire regulations will permit a maximum of
500 people at the dance. Domain? Range?
(d) Determine the cost of the dance if 400 people
attend.
(e) How many people can attend if the council can
only afford to spend $2350?
(f) What quantity does the slope of the line
represent?
(g) Determine the y-intercept. What meaning does the intercept have?
3. In order to determine the gasoline consumption of a new car, the company representative filled the tank with gas and proceeded to drive around the test track at a constant speed. After driving 100 km, 80 L of gas remained. After 300 km, 40 L of gas remained.
(a) Plot a graph of this relation.
(b) Find an equation of this relation expressing litres of gas
remaining in terms of km driven.
(c) What quantity does the slope of the relation represent?
(d) How many litres of gas remain after 400 km of driving?
(e) What is the capacity of the gas tank in litres?
(f) How many km could you expect to drive on a full tank of gas?
(g) How many litres of gas will the car use in going 350 km?
(h) State the domain and range of this relation
4. The length of a steel beam is a linear function of the temperature. When the temperature is 40 degrees F a certain steel beam is 50 feet long and at a temperature of 95 degrees F the same beam is 50.12 feet long.
(a) Determine the equation of this relation.
(b) What quantity does the slope of the line represent?
(c) What is the length of the beam at 70 degrees F.
(d) What is the y-intercept? What does it represent?
(e) At what temperature is the beam 49.9 feet long?
5. Thrifty charges $42.95 a day plus $0.29 a mile for the rental of a car.
(a) Write an equation showing the relationship between the
daily cost to rent a car and the number of miles you drive.
(b) Write the slope and the y-intercept of the line, and briefly
explain the meaning of each in the car rental context.
(c) If you have a daily budget of $75, what is the maximum
distance you can drive each day and stay within your
budget?
(d) What would be the total cost of using the rental car for 3
days and driving the car 304.8 miles?
6. Peter drove from his mother’s home to his sister’s home. After driving for 20 minutes he was 62 miles away from his sister’s home and after driving for 32 minutes he was only 38 miles away. The time driving and the distance away from his sister’s home form a linear relationship.
(a) What is the independent variable? What is the dependent variable?
(b) What are the two data values?
(c) Draw a graph to represent this problem. Label the axis appropriately.
(d) Write an equation expressing distance in terms of time driving.
(e) What is the slope and what is its meaning in this problem?
(f) What is the time-intercept and what does it represent?
(g) What is the distance-intercept and what does it represent?
(h) How far is Peter from his sister’s home after he had been driving for 35 minutes?
7. Players on the school soccer team are selling candles to raise money for an upcoming trip. Each player has 24 candles to sell. If a player sells 4 candles a profit of $30 is made. If he sells 12 candles a profit of $70 is made. The profit and the number of candles sold form a linear relation.
(a) State the dependent and the independent variables.
(b) Determine an equation to model this situation.
(c) What is the slope and what does it mean in this problem?
(d) Find the profit-intercept and explain what it represents.
(e) Calculate the maximum profit that a player can make.
(f) Write a suitable domain and range.
(g) If a player makes a profit of $90, how many candles did he sell?
(h) Is this data continuous or discrete? Justify your answer.