5. John and Daphne are saving for their daughter Ellen's college education. Ellen just turned 10 (t=0), and she will be entering college 8 years from now (at t=8). College tution and expenses at State U. are currently $14,500 a year, but they are expected to increase at a rate of 3.5% a year. Ellen should graduate in 4 years-if she takes longer or wants to go to graduate school, she will be on her own. Tution and other costs will be due at the begining of each school year (at t=8,9,10, and 11).

So far,John and Daphne hace accumulated $15,000 in their college savings account (at t=0). Their long-run financial olan is to add an additional $5,000 in each of the next 4 years(at t=1,2,3, and 4). Then they plan to make 3 equal annual contributions in each of the following years, t=5,6, and 7. They expect their investment account to earn 9%. How large must the annual payments at t= 5,6, and 7 be to cover Ellen's anticipated college costs?

a. $1,965.21
b. $2,068.64
c. $2,177.51
d. $2,292.12
e. $2,412.76

2412

2412.76

. John and Daphne are saving for their daughter Ellen's college education. Ellen just turned 10 (t=0), and she will be entering college 8 years from now (at t=8). College tution and expenses at State U. are currently $14,500 a year, but they are expected to increase at a rate of 3.5% a year. Ellen should graduate in 4 years-if she takes longer or wants to go to graduate school, she will be on her own. Tution and other costs will be due at the begining of each school year (at t=8,9,10, and 11).

So far,John and Daphne hace accumulated $15,000 in their college savings account (at t=0). Their long-run financial olan is to add an additional $5,000 in each of the next 4 years(at t=1,2,3, and 4). Then they plan to make 3 equal annual contributions in each of the following years, t=5,6, and 7. They expect their investment account to earn 9%. How large must the annual payments at t= 5,6, and 7 be to cover Ellen's anticipated college costs?

a. $1,965.21
b. $2,068.64
c. $2,177.51
d. $2,292.12
e. $2,412.76

5. John and Daphne are saving for their daughter Ellen's college education. Ellen just turned 10 at (t = 0), and she will be entering college 8 years from now (at t = 8). College tuition and expenses at State U. are currently $14,500 a year, but they are expected to increase at a rate of 3.5% a year. Conclusion

To determine the size of the annual payments at t=5, 6, and 7, we need to calculate the future value (FV) of John and Daphne's college savings account and set it equal to the present value (PV) of Ellen's anticipated college costs.

First, let's calculate the future value (FV) of John and Daphne's college savings account. Since they will be making equal annual contributions for the next 4 years (t=1, 2, 3, and 4), we can use the formula for the future value of an ordinary annuity:

FV = Pmt × [(1 + r)^(n-1) / r]

Where:
Pmt = Annual payment
r = Interest rate
n = Number of periods

Using the given information:
Pmt = $5,000
r = 9% or 0.09
n = 4

FV = $5,000 × [(1 + 0.09)^(4-1) / 0.09]
FV = $5,000 × [(1.09)^3 / 0.09]
FV = $5,000 × [1.2950305 / 0.09]
FV ≈ $71,278.17

Next, we need to calculate the present value (PV) of Ellen's anticipated college costs. Since tuition and expenses will increase by 3.5% per year, we can use the formula for the present value of a growing annuity:

PV = Pmt x [(1 - (1 + g)^(-n)) / (r - g)]

Where:
Pmt = Annual payment (tuition and expenses)
r = Interest rate
n = Number of periods (4 years of college)
g = Growth rate (3.5%)

Using the given information:
Pmt = $14,500
r = 9% or 0.09
n = 4
g = 3.5% or 0.035

PV = $14,500 × [(1 - (1 + 0.035)^(-4)) / (0.09 - 0.035)]
PV ≈ $49,927.49

Finally, we need to find the annual payments at t=5, 6, and 7 that will cover Ellen's anticipated college costs. We can use the formula for the present value of an ordinary annuity:

PV = Pmt × [(1 - (1 + r)^(-n)) / r]

Using the given information:
PV = $49,927.49
r = 9% or 0.09
n = 3

$49,927.49 = Pmt × [(1 - (1 + 0.09)^(-3)) / 0.09]
$49,927.49 ≈ Pmt × (1 - 0.77218341) / 0.09
$49,927.49 ≈ Pmt × (0.22781659) / 0.09
$49,927.49 ≈ Pmt × 2.53129

Pmt ≈ $49,927.49 / 2.53129
Pmt ≈ $19,681.17

Therefore, the annual payments at t=5, 6, and 7 must be approximately $19,681.17 to cover Ellen's anticipated college costs.

However, none of the options provided match this amount exactly.