Linear functions are highly used throughout mathematics and are therefore important to understand.
A set of problems involving linear functions, along with detailed solutions, are presented. The problems are designed with emphasis on the meaning of the slope and the y intercept.
f is a linear function. Values of x and f(x) are given in the table below; complete the table.
Solution to Problem 1:
A family of linear functions is given by
Solution to Problem 2:
A high school had 1200 students enrolled in 2003 and 1500 students in 2006. If the student population P ; grows as a linear function of time t, where t is the number of years after 2003.
a) How many students will be enrolled in the school in 2010?
b) Find a linear function that relates the student population to the time t.
Solution to Problem 3:
a) The given information may be written as ordered pairs (t , P). The year 2003 correspond to t = 0 and the year 2006 corresponds to t = 3, hence the 2 ordered pairs
(0, 1200) and (3, 1500)
Since the population grows linearly with the time t, we use the two ordered pairs to find the slope m of the graph of P as follows
m = (1500 - 1200) / (6 - 3) = 100 students / year
The slope m = 100 means that the students population grows by 100 students every year. From 2003 to 2010 there are 7 years and the students population in 2010 will be
P(2010) = P(2003) + 7 * 100 = 1200 + 700 = 1900 students.
b) We know the slope and two points, we may use the point slope form to find an equation for the population P as a function of t as follows
P - P1 = m (t - t1)
P - 1200 = 100 (t - 0)
P = 100 t + 1200
The graph shown below is that of the linear function that relates the value V (in $) of a car to its age t, where t is the number of years after 2000.
.a) Find the slope and interpret it.
The cost of producing x tools by a company is given by
A 500-liter tank full of oil is being drained at the constant rate of 20 liters par minute.
a) Write a linear function V for the number of liters in the tank after t minutes (assuming that the drainage started at t = 0).
b) Find the V and the t intercepts and interpret them.
e) How many liters are in the tank after 11 minutes and 45 seconds?
Solution to Problem 6:
After each minute the amount of oil in the tank deceases by 20 liters. After t minutes, the amount of oil in the tank decreases by 20*t liters. Hence if at the start there 500 liters, after t minute the amount V of oil left in the tank is given by
V = 500 - 20 t
b) To find the V intercept, set t = 0 in the equation V = 500 - 20 t.
V = 500 liters : it is the amount of oil at the start of the drainage.
To find the t intercept, set V = 0 in the equation V = 500 - 20 t and solve for t.
0 = 500 - 20 t
t = 500 / 20 = 25 minutes : it is the total time it takes to drain the 500 liters of oil.
c) Convert 11 minutes 45 seconds in decimal form.
t = 11 minutes 45 seconds = 11.75 minutes
Calculate V at t = 11.75 minutes.
V(11.75) = 500 - 20*11.75 = 265 liters are in the tank after 11 minutes 45 seconds of drainage.
A 50-meter by 70-meter rectangular garden is surrounded by a walkway of constant width x meters.
A driver starts a journey with 25 gallons in the tank of his car. The car burns 5 gallons for every 100 miles. Assuming that the amount of gasoline in the tank decreases linearly,
a) write a linear function that relates the number of gallons G left in the tank after a journey of x miles.
b) What is the value and meaning of the slope of the graph of G?
c) What is the value and meaning of the x intercept?
Solution to Problem 8:
a) If 5 gallons are burnt for 100 miles then (5 / 100) gallons are burnt for 1 mile. Hence for x miles, x * (5 / 100) gallons are burnt. G is then equal to the initial amount of gasoline decreased by the amount gasoline burnt by the car. Hence
G = 25 - (5 / 100) x
b) The slope of G is equal to 5 / 1000 and it represent the amount of gasoline burnt for a distance of 1 mile.
c) To find the x intercept, we set G = 0 and solve for x.
25 - (5 / 100) x = 0
x = 500 miles : it is the distance x for which all 25 gallons of gasoline will be burnt.
A rectangular wire frame has one of its dimensions moving at the rate of 0.5 cm / second. Its width is constant and equal to 4 cm. If at t = 0 the length of the rectangle is 10 cm,