4. The yearly cost T, in thousands of dollars, of tuition and required fees at a private college (includes two- and four-year schools and does not include room and board) can be approximated by
T= 3/5n + 5
where n is the number of years since 1985. That is, corresponds to 1985, corresponds to 1990, and so on. Show your work and explain each answer.
Source: Statistical Abstract of the United States, 2003
a. Find the cost of tuition in 1985, 1996, 2000, and 2004.
b. Graph the equation and then use the graph to estimate the cost of tuition in 2005. Use the drawing tools to draw your line on the graph.
c. Predict the year in which the cost of tuition will be $23,000. Explain your answer.
1985 , --- t = 0
1996 , --- t = 11
etc.
b) you will have to do the graph, hard to show graphs on here
c) set T = 23
23 = (3/5)n + 5
18 = (3/5)n
90 = 3n
n= 30 years from 1985 or in 2075
a. To find the cost of tuition in different years, we need to substitute the corresponding values of n into the equation T = (3/5)n + 5.
For 1985:
n = 1985 - 1985 = 0
T = (3/5)(0) + 5
T = 5
So the cost of tuition in 1985 was $5,000.
For 1996:
n = 1996 - 1985 = 11
T = (3/5)(11) + 5
T = 33/5 + 5
T = 6.6 + 5
T = 11.6
So the cost of tuition in 1996 was $11,600.
For 2000:
n = 2000 - 1985 = 15
T = (3/5)(15) + 5
T = 45/5 + 5
T = 9 + 5
T = 14
So the cost of tuition in 2000 was $14,000.
For 2004:
n = 2004 - 1985 = 19
T = (3/5)(19) + 5
T = 57/5 + 5
T = 11.4 + 5
T = 16.4
So the cost of tuition in 2004 was $16,400.
b. To graph the equation, we can plot the values of n on the x-axis and the corresponding values of T on the y-axis.
The graph will be a straight line with a slope of 3/5 and a y-intercept of 5.
To estimate the cost of tuition in 2005, we can simply substitute n = 2005 - 1985 = 20 into the equation and solve.
T = (3/5)(20) + 5
T = 60/5 + 5
T = 12 + 5
T = 17
So the estimated cost of tuition in 2005 is $17,000.
c. To predict the year in which the cost of tuition will be $23,000, we need to solve the equation T = (3/5)n + 5 for n.
(3/5)n + 5 = 23
(3/5)n = 23 - 5
(3/5)n = 18
To isolate n, we can multiply both sides by 5/3.
n = 18 * (5/3)
n = 30
So the cost of tuition will be $23,000 in the year 2015 (1985 + 30).