Functions Word Problem

Question

Jerry plans to begin saving money for a trip by putting $1 in a savings account the first month and then adding enough to double the amount in the account each following month.

After he finishes contributing to the account, he will withdraw $500 to make a down payment on the trip. Create a function to show how much money will be in the account if he finishes contributing to the account after t months.

Type your response here: I think the first answer is S(t) = 2^t - 1 .


Jerry plans to keep up this saving pattern until he can no longer afford it. At that time, he hopes to have enough in his account for his trip. How much will be in the account, after making the down payment, if the most he will be able to contribute in a month is $1,200?

Type your response here:


Jerry’s friend Brandon is planning to start saving for a trip at the same time as Jerry. The graph shows the balance in Brandon’s account over the first few months of saving. Whose account will have the greatest balance after the first five months of savings, Brandon’s or Jerry’s?



Type your response here:


How long will it take for the person with the smaller account balance after five months to have the greater balance?

Type your response here:


Brandon will also need to withdraw $500 to make a down payment on the trip. How much will be in Brandon’s account, after making the down payment, if he contributes for the same amount of time as Jerry? (Review your calculation in part b to determine how long Jerry continued his savings plan.)

Type your response here:

Answer 1

To check your equation lets write out some terms of the sequence

 

1st month = 1,  total in account is 1

 

2nd month 1 + 1 = 2,  total in account is 2

 

3rd month 2 + 2 = 4,  total in account is 4

 

4th month 4 + 4 = 8, total in account is 8

 

5th month 8 + 8 = 16,  total in account is 16

 

So the sequence we have is

 

{1, 1+1, 2 + 2, 4 + 4, 8 + 8,...}→{1, 2, 4, 8, 16,...}

 

As I see it this is the sequence of partial sums so he does not have 1 + 2 =3 after two months.  He did not deposit two dollars for the second month.  He only puts enough money in the account to double what was already there.  So the summation equation is 

 

s_t = 2^{(t - 1)},  t ≥ 1

 

Set this equation less than or equal to 1200 and solve for t

 

1200 ≤ 2^{(t - 1)}

 

Take the log of each side

 

log(1200) ≤ log(2^{(t - 1)})

 

Use the power rule

 

log(1200) ≤ (t - 1)log(2)

 

divide each side by log(2) and add 1

log(1200)/log(2) ≤ t - 1

 

log(1200)/log(2) + 1 ≤ t

 

t ≥ 11.23, rounding up gives t = 12

 

so he can contribute to the account for 12 months

 

At 12 months he has

 

s₁₂ = 2^¹¹ = 2048,  1024 in account and 1024 that he deposits

 

What left in the account is 2048 - 500 = 1548