Suppose $$ \$ 4,800 $$ is deposited into an account which has an annual percentage rate of $$ 7.27 \% $$ per year. Assume it remains in the account for 39 years and no additional money is added to the account other than interest. (a) Complete the boxes below to write an expression for the amount (in dollars) at year $$ x=39 $$, of assuming interest is compounded annually. Do not round any values. You can enter arithmetic expressions (containing $$ +,-{ }^{*}, / $$, or $$ { }^{\wedge} $$ ) in any of these boxes. Number Number - Number ) What is the value in year $$ x=39 $$ of an investment of $$ \$ 4,800 $$ dollars which pays $$ 7.27 \% $$ compounded annually? \$ $$ \square $$ Number (Round to the nearest 0.01 dollars) (b) Complete the boxes below to write an expression for the amount (in dollars) at year $$ x=39 $$, assuming interest is compounded daily ( 365 times per year). Do not round any values. You can enter arithmetic expressions (containing $$ +,-,{ }^{*}, / $$, or $$ \wedge $$ ) in any of these boxes. Number Number Number ) What is the value in year $$ x=39 $$ of an investment of $$ \$ 4,800 $$ dollars which pays $$ 7.27 \% $$ compounded daily? \$ $$ \square $$ Number (Round to the nearest 0.01 dollars) (c) Complete the boxes below to write an expression for the amount (in dollars) at year $$ x=39 $$, assuming interest is compounded continuously. Do not round any values. You can enter arithmetic expressions (containing $$ +,-,{ }^{*}, / $$, or $$ \wedge $$ ) in any of these boxes. Number Number - $$ e $$ What is the value in year $$ x=39 $$ of an investment of $$ \$ 4,800 $$ dollars which pays $$ 7.27 \% $$ compounded continuously?

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To find the amount in the account after 39 years with different compounding methods, we can use the formula for compound interest:

$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$

where:
- $$ A $$ is the amount in the account after $$ t $$ years,
- $$ P $$ is the principal amount (initial deposit),
- $$ r $$ is the annual interest rate (in decimal form),
- $$ n $$ is the number of times interest is compounded per year,
- $$ t $$ is the time in years.

Given:
- Principal amount $$ P = \$4,800 $$,
- Annual interest rate $$ r = 7.27\% = 0.0727 $$,
- Time $$ t = 39 $$ years.

(a) For annual compounding:
$$ A = 4800 \left(1 + \frac{0.0727}{1}\right)^{1 \cdot 39} $$

(b) For daily compounding:
$$ A = 4800 \left(1 + \frac{0.0727}{365}\right)^{365 \cdot 39} $$

(c) For continuous compounding:
$$ A = 4800 e^{0.0727 \cdot 39} $$

Now, let's calculate the values for each compounding method.
Calculate the value by following steps:
- step0: Calculate:
  $$4800\left(1+0.0727\right)^{39}$$
- step1: Add the numbers:
  $$4800\times 1.0727^{39}$$
- step2: Convert the expressions:
  $$4800\left(\frac{10727}{10000}\right)^{39}$$
- step3: Simplify:
  $$4800\times \frac{10727^{39}}{10000^{39}}$$
- step4: Rewrite the expression:
  $$400\times 12\times \frac{10727^{39}}{10000^{39}}$$
- step5: Rewrite the expression:
  $$400\times 12\times \frac{10727^{39}}{400^{39}\times 25^{39}}$$
- step6: Reduce the numbers:
  $$12\times \frac{10727^{39}}{400^{38}\times 25^{39}}$$
- step7: Rewrite the expression:
  $$4\times 3\times \frac{10727^{39}}{400^{38}\times 25^{39}}$$
- step8: Rewrite the expression:
  $$4\times 3\times \frac{10727^{39}}{16^{38}\times 25^{38}\times 25^{39}}$$
- step9: Rewrite the expression:
  $$4\times 3\times \frac{10727^{39}}{4^{76}\times 25^{38}\times 25^{39}}$$
- step10: Reduce the numbers:
  $$3\times \frac{10727^{39}}{4^{75}\times 25^{38}\times 25^{39}}$$
- step11: Multiply the fractions:
  $$\frac{3\times 10727^{39}}{4^{75}\times 25^{77}}$$
Calculate or simplify the expression $$ 4800*(1+0.0727/365)^{365*39} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$4800\left(1+\frac{0.0727}{365}\right)^{365\times 39}$$
- step1: Divide the terms:
  $$4800\left(1+\frac{727}{3650000}\right)^{365\times 39}$$
- step2: Add the numbers:
  $$4800\left(\frac{3650727}{3650000}\right)^{365\times 39}$$
- step3: Multiply the numbers:
  $$4800\left(\frac{3650727}{3650000}\right)^{14235}$$
- step4: Simplify:
  $$4800\times \frac{3650727^{14235}}{3650000^{14235}}$$
- step5: Rewrite the expression:
  $$400\times 12\times \frac{3650727^{14235}}{3650000^{14235}}$$
- step6: Rewrite the expression:
  $$400\times 12\times \frac{3650727^{14235}}{400^{14235}\times 9125^{14235}}$$
- step7: Reduce the numbers:
  $$12\times \frac{3650727^{14235}}{400^{14234}\times 9125^{14235}}$$
- step8: Rewrite the expression:
  $$4\times 3\times \frac{3650727^{14235}}{400^{14234}\times 9125^{14235}}$$
- step9: Rewrite the expression:
  $$4\times 3\times \frac{3650727^{14235}}{16^{14234}\times 25^{14234}\times 9125^{14235}}$$
- step10: Rewrite the expression:
  $$4\times 3\times \frac{3650727^{14235}}{4^{28468}\times 25^{14234}\times 9125^{14235}}$$
- step11: Reduce the numbers:
  $$3\times \frac{3650727^{14235}}{4^{28467}\times 25^{14234}\times 9125^{14235}}$$
- step12: Multiply the fractions:
  $$\frac{3\times 3650727^{14235}}{4^{28467}\times 25^{14234}\times 9125^{14235}}$$
Calculate or simplify the expression $$ 4800*e^(0.0727*39) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$4800e^{0.0727\times 39}$$
- step1: Multiply the numbers:
  $$4800e^{2.8353}$$
- step2: Convert the expressions:
  $$4800e^{\frac{28353}{10000}}$$
(a) The amount in the account after 39 years with annual compounding is approximately $74,112.96.

(b) The amount in the account after 39 years with daily compounding is approximately $81,747.37.

(c) The amount in the account after 39 years with continuous compounding is approximately $81,770.45.
(a) The amount after 39 years with annual compounding is approximately \$74,112.96. (b) The amount after 39 years with daily compounding is approximately \$81,747.37. (c) The amount after 39 years with continuous compounding is approximately \$81,770.45.

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