According to the College Board the average cost of tuition i

According to the College Board, the average cost of tuition (including fees, room and board) at private four-year nonprofit colleges has risen from $24, 071 in 2003 to $30, 094 in 2013. What will the average cost of tuition be in 2023 assuming linear growth? How much does the tuition increase per year? What will the average cost of tuition be in 2023 assuming exponential growth? By what percent does the tuition increase per year? Regardless of which model we use, the cost of tuition is definitely increasing. Now, if the cost grows exponentially, is it concave up or concave down? Support your answer by computing the average rate of change from 2013 to 2023 and comparing it with the average rate of change from 2003 to 2013.

Solution

a)in 2003, t =0, in 2013, t =2013-2003=10

linear equation joining points (0,24071),(10,30094) is

y-24071=[(30094-24071)/(10-0)](t-0)

y-24071=602.3t

y=24071+602.3t

in the year 2023, t =2023-2003=20

y=24071+(602.3*20)

y=36117

(t,y)=(20,36117)

average cost of tuition in 2023 is 36117 dollars

tuition increase 602.3 dollars per year

b)general form of expenential equation is y =ab^t

at (0,24071)

24071=a*b⁰

24071=a*1

a=24071

y =24071b^t

at (10,30094)

30094 =24071*b¹⁰

=>b =1.022583

y =24071*1.02258^t

in 2023, t =20

y =24071*1.02258²⁰

y=37624 dollars

(t,y)=(20,37624)

growth factor =1.022583

growth rate =0.022583

percent rate =2.2583 %

tution increase at 2.26 % per year

c) average rate of change from 2013 to 2023 i.e, between (10,30094),(20,37624)

=[37624-30094]/(20-10)

=753

average rate of change from 2003 to 2013 i.e, between (0,24071),(10,30094)

=[30094-24071]/(10-0)

=602.3

average cost increased

so if cost grows exponentially it is concave up