samuelmeier187
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If 26500 dollars is invested at an interest rate of 6 percent per year, find the value of the investment at the end of 5 years for the following compounding methods, to the nearest cent.

(a) Annual: $
(b) Semiannual: $
(c) Monthly: $
(d) Daily: $



Answer :

msm555

Answer:

(a) Annual Compounding: $35462.98

(b) Semiannual Compounding: $35613.78

(c) Monthly Compounding: $35744.53

(d) Daily Compounding: $35770.38

Step-by-step explanation:

Let's calculate the value of the investment at the end of 5 years using the different compounding methods, where:

  • [tex]\sf P = \$26500 [/tex] (principal amount)
  • [tex]\sf r = 6\% [/tex] per year (interest rate)
  • [tex]\sf t = 5 [/tex] years (time)

(a) Annual Compounding:

We can use formula:

[tex]\large\boxed{\boxed{\sf \textsf{Compound Amount} = P \left(1 + \dfrac{r}{100}\right)^t}} [/tex]

Where

  • P is the principal amount.
  • r is the rate of interest.
  • t is time for investment.

Substitute the value and simplify:

[tex]\begin{aligned} \textsf{Compound Amount} &= 26500 \left(1 + \dfrac{6}{100}\right)^5\\\\ & = 26500 \left(1 + 0.06\right)^5\\\\ & = 26500 \times (1.06)^5 \\\\ & = 26500 \times 1.3382255776 \\\\ & = 35462.9778064 \\\\ & = 35462.98 \textsf{(in nearest cent)} \end{aligned} [/tex]

[tex]\dotfill[/tex]

(b) Semiannual Compounding:

We can use formula:

[tex]\large\boxed{\boxed{\textsf{Compound Amount} = P \left(1 + \dfrac{r}{200}\right)^{2t}}} [/tex]

Where

  • P is the principal amount.
  • r is the rate of interest.
  • t is time for investment.

Substitute the value and simplify:

[tex]\begin{aligned} \textsf{Compound Amount} & = 26500 \left(1 + \dfrac{6}{200}\right)^{2 \times 5} \\\\ & = 26500 \left(1 + 0.03\right)^{10} \\\\ & = 26500 \times (1.03)^{10} \\\\ & = 26500 \times 1.34391637934412192049 \\\\ & = 35613.784052619230892985 \\\\ & = 35613.78 \textsf{(in nearest cent)}\end{aligned} [/tex]

[tex]\dotfill[/tex]

(c) Monthly Compounding:

We can use formula:

[tex]\large\boxed{\boxed{\sf \textsf{Compound Amount} = P \left(1 + \dfrac{r}{1200}\right)^{12t} }}[/tex]

Where

  • P is the principal amount.
  • r is the rate of interest.
  • t is time for investment.

Substitute the value and simplify:

[tex]\begin{aligned} \textsf{Compound Amount} & = 26500 \left(1 + \dfrac{6}{1200}\right)^{12 \times 5} \\\\ & = 26500 \left(1 + 0.005\right)^{60} \\\\ & = 26500 \times (1.005)^{60} \\\\ & = 26500 \times 1.3488501525493 \\\\ & = 35744.529042556 \\\\ & = 35744.53 \textsf{(in nearest cent)}\end{aligned} [/tex]

[tex]\dotfill[/tex]

(d) Daily Compounding:

We can use formula:

[tex]\large\boxed{\boxed{\sf \textsf{Compound Amount} = P \left(1 + \dfrac{r}{36500}\right)^{365t}}} [/tex]

Where

  • P is the principal amount.
  • r is the rate of interest.
  • t is time for investment.

Substitute the value and simplify:

[tex]\begin{aligned} \textsf{Compound Amount} & = 26500 \left(1 + \dfrac{6}{36500}\right)^{365 \times 5} \\\\ & = 26500 \left(1 + 0.0001643835616\right)^{1825} \\\\ & = 26500 \times (1.00016438356)^{1825} \\\\ & = 26500 \times 1.3498255274436 \\\\ & = 35770.376477255 \\\\ & = 35770.38 \textsf{(in nearest cent)} \end{aligned} [/tex]

Therefore, the value of the investment at the end of 5 years for each compounding method (rounded to the nearest cent) are:

(a) Annual Compounding: $35462.98

(b) Semiannual Compounding: $35613.78

(c) Monthly Compounding: $35744.53

(d) Daily Compounding: $35770.38

semsee45

Answer:

(a)  $35,462.98

(b)  $35,613.78

(c)  $35,744.53

(d)  $35,770.38

Step-by-step explanation:

To find the value of the investment at the end of 5 years for each compounding method, we can use the formula for compound interest:

[tex]\boxed{\begin{array}{l}\underline{\textsf{Compound Interest Formula}}\\\\A=P\left(1+\dfrac{r}{n}\right)^{nt}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$A$ is the final amount.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount.}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$n$ is the number of times interest is applied per year.}\\\phantom{ww}\bullet\;\;\textsf{$t$ is the time (in years).}\end{array}}[/tex]

In this case:

  • P = $26,500
  • r = 6% = 0.06
  • t = 5 years

Substitute these values into the formula:

[tex]A=26500\left(1+\dfrac{0.06}{n}\right)^{5n}[/tex]

Now, calculate for each compounding method by substituting the corresponding value of n into the equation.

(a) Annual compounding: n = 1

[tex]A=26500\left(1+\dfrac{0.06}{1}\right)^{5\cdot 1}\\\\\\A=26500\left(1.06\right)^{5}\\\\A=35462.9778064\\\\A=\$35462.98[/tex]

(b) Semi-annual compounding: n = 2

[tex]A=26500\left(1+\dfrac{0.06}{2}\right)^{5\cdot 2}\\\\\\A=26500\left(1.03\right)^{10}\\\\A=35,613.7840526...\\\\A=\$35613.78[/tex]

(c) Monthly compounding: n = 12

[tex]A=26500\left(1+\dfrac{0.06}{12}\right)^{5\cdot 12}\\\\\\A=26500\left(1.005\right)^{60}\\\\A=35,744.52904255...\\\\A=\$35744.53[/tex]

(d) Daily compounding: n = 365

[tex]A=26500\left(1+\dfrac{0.06}{365}\right)^{5\cdot 365}\\\\\\A=26500\left(1.00016438...\right)^{1825}\\\\A=35770.3764772...\\\\A=\$35770.38[/tex]

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