Problem 1 :
Suppose that a bike rents for $4 plus $1.50 per hour. Write an equation in slope-intercept form that models this situation.
Solution :
Let y be the total cost for renting bike for x hours.
From the given information, we have
y = 1.5x + 4
Problem 2 :
In order to become a member of the library-all-star-members club, there is a $40 sign-up fee and a $2 monthly fee. Write an equation in slope-intercept form that models this situation. Use the equation to find the total cost of being an all-star library member for 19 months.
Solution :
Let y be the total cost of being an all-star library member for x months.
From the given information, we have
y = 2x + 40
To find the total cost of being an all-star library member for 19 months, substitute x = 19 into the above equation.
y = 2(19) + 40
y = 38 + 40
y = 78
The total cost of being an all-star library member for 19 months is $78.
Problem 3 :
A taxi service charges a fixed rate of $30 and $3 per mile. Write an equation in slope-intercept form that models this situation. Use the equation to find the total cost for an 18-miles trip.
Solution :
Let y be the total cost for travelling x miles.
From the given information, we have
y = 3x + 30
To find the total cost for an 18-miles trip, substitute x = 18 into the above equation.
y = 3(18) + 30
y = 54 + 30
y = 84
The total cost for an 18-miles trip is $84.
Problem 4 :
The U.S. Bureau of the Census predicted that the population of Florida would be about 17.4 million in 2010 and then would increase by about 0.22 million per year until 2015. Which of the following linear models predicts the population, y, of Florida (in millions) in terms of x, the number of years since 2010?
(A) y = 17.4x + 0.22
(B) y = -0.22x + 17.4
(C) y = 0.22x + 17.4
(D) y = -17.4x + 0.22
Solution :
If we consider 2010 as starting year, the corresponding values we have for 2010 and 2015 are 0 and 5 respectively.
2010 ----> 0
2015 ----> 5
From the given information, we can have the following points.
(0, 17.4) and (5, 0.22)
It is given that the population would increase by about 0.22 million per year. So, the slope is 0.22.
m = 0.22
Equation which models the given situation is
y = mx + b
y = 0.22x + b
To get the value of b, we can substitute one of the two points (0, 17.4) or (5, 0.22) into the equation above.
Substitute (x, y) = (0, 17.4).
17.4 = 0.22(0) + b
17.4 = 0 + b
17.4 = b
Equation in slope-intercept form :
y = 0.22x + 17.4
Therefore, the correct answer is option is option (C).
Problem 5 :
In 2005, 120 students at Lincoln High School had smart phones. By 2010, 345 students in the same school had smart phones. Write a linear model which predicts the number of students having smart phones and estimate the number of students having smart phones in 2018.
Solution :
Let x represent the number of years since 2005 and y represent the number students having smart phones in a particular year.
If we consider 2005 as starting year, the corresponding values we have for 2005, 2010 and 2018 are 0, 5 and 8 respectively.
2010 ----> 0
2015 ----> 5
2018 ----> 8
From the given information, we can have the following points.
(0, 120) and (5, 345)
Formula to find the slope of a line joining two points :
Substitute (x_{1}, y_{1}) = (0, 120) and (x_{2}, y_{2}) = (5, 345).
m = 45
Equation in slope-intercept form :
y = mx + b
y = 45x + b
To get the value of b, substitute (x, y) = (0, 120).
120 = 45(0) + b
120 = 0 + b
120 = b
The linear model which predicts the number of students having smart phones :
y = 45x + 120
To estimate the number of students having smart phones in 2018, substitute x = 8 into the above equation.
y = 45(8) + 120
y = 360 + 120
y = 480
It is estimated that there would be 480 students having smart phone in 2008.
Problem 6 :
A cab service charges a fixed rate of $50 for a trip of 10 miles or less. It is charged $4 for each additional mile apart from 10 miles. Write an equation in slope-intercept form which gives the total cost for a trip of more than 10 miles. Use the equation to find the total cost of a 25-miles trip.
Solution :
Let y be the total cost of a trip of x miles, where x > 10.
Then, (x - 10) represents each additional mile travelled apart from 10 miles.
From the given information, we have
y = 50 + 4(x - 10)
y = 50 + 4x - 40
y = 4x + 10
To find the total cost of a 25-miles trip, substitute x = 25 into the above equation.
y = 4(25) + 10
y = 100 + 10
y = 110
The total cost of a 25-miles trip is $110.
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