Part 4 of 4 When Tanya was born, her family put $$ \$ 2,500 $$ into an account for her that earns $$ 5.5 \% $$ annually. Her family plans to use the money to purchase a trip for her when she graduates from high school. Complete parts a through d below. Package A - London - Paris \$5,000 Package B - London - Paris - Madrid Package C - London - Paris - Madrid - Rome d. She decides to wait to purchase the vacation package until she graduates from college, at age 22 . She hopes that she will be able to afford Package D which includes trips to London, Paris, Madrid, Rome, and Athens. Package D costs $$ \$ 9,500 $$. Will she have enough money? If not, how much more money does she need? Select all that apply and fill in the answer boxes to complete your choice. (Round to two decimal places as needed) A. She will have $$ \$ \square $$ $$ \square $$ in the account, so she will have enough money. B. She will have $$ \$ $$ $$ \square $$ in the account, so she will need \$ $$ \square $$ more. Clear all Check answer Video Textbook Get more help -

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UpStudy AI · Accepted Answer

To determine if Tanya will have enough money to purchase Package D when she graduates from college, we need to calculate the total amount of money she will have in the account at that time.

Given:
- Initial amount in the account: $2,500
- Annual interest rate: 5.5%
- Time period: 22 years (from birth to graduation from college)

First, let's calculate the total amount of money Tanya will have in the account after 22 years using the formula for compound interest:

$$ A = P(1 + r)^n $$

where:
- $$ A $$ is the total amount of money after $$ n $$ years
- $$ P $$ is the initial amount of money
- $$ r $$ is the annual interest rate
- $$ n $$ is the number of years

Substitute the given values into the formula:

$$ A = 2500(1 + 0.055)^{22} $$

Now, let's calculate the total amount of money Tanya will have in the account after 22 years.
Calculate the value by following steps:
- step0: Calculate:
  $$2500\left(1+0.055\right)^{22}$$
- step1: Add the numbers:
  $$2500\times 1.055^{22}$$
- step2: Convert the expressions:
  $$2500\left(\frac{211}{200}\right)^{22}$$
- step3: Simplify:
  $$2500\times \frac{211^{22}}{200^{22}}$$
- step4: Rewrite the expression:
  $$100\times 25\times \frac{211^{22}}{200^{22}}$$
- step5: Rewrite the expression:
  $$100\times 25\times \frac{211^{22}}{100^{22}\times 2^{22}}$$
- step6: Reduce the numbers:
  $$25\times \frac{211^{22}}{100^{21}\times 2^{22}}$$
- step7: Rewrite the expression:
  $$25\times \frac{211^{22}}{25^{21}\times 4^{21}\times 2^{22}}$$
- step8: Reduce the numbers:
  $$1\times \frac{211^{22}}{25^{20}\times 4^{21}\times 2^{22}}$$
- step9: Multiply the fractions:
  $$\frac{211^{22}}{25^{20}\times 2^{64}}$$
The total amount of money Tanya will have in the account after 22 years is approximately $8,118.84.

Now, let's compare this amount to the cost of Package D, which is $9,500. If the amount in the account is less than the cost of Package D, Tanya will need to save more money to afford the package.

Since $8,118.84 is less than $9,500, Tanya will need to save more money to afford Package D.

Therefore, the correct answer is:
B. She will have $8,118.84 in the account, so she will need $1,381.16 more.
She will have $8,118.84 in the account, so she will need $1,381.16 more.

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