Math / Algebra

Questioninvested at $3.5 \%$ interest for 8 yr under the following compounding options. Round answers in the second column to the nearest whole number. Round answers in the last column to the nearest cent. \begin{tabular}{|l|l|c|c|} \hline & Compounding Option & $n$ Value & Result \\ \hline (a) & Annually & $n=\square$ & $\$ \square$ \\ \hline (b) & Quarterly & $n=\square$ & $\$ \square$ \\ \hline (c) & Monthly & $n=\square$ & $\$ \square$ \\ \hline (d) & Daily & $n=365$ & $\$ \square$ \\ \hline (e) & Continuously & Not Applicable & $\$ \square$ \\ \hline \end{tabular} Check Save For Later Submit Assignment

Studdy Solution

STEP 1

What is this asking? We're figuring out how much money we'll have after 8 years if we invest $\$9000$ at a $\$3.5\%$ interest rate, compounded in different ways. Watch out! Don't mix up the different compounding periods!
Also, remember that continuous compounding is a little different.

STEP 2

1. Calculate Annually
2. Calculate Quarterly
3. Calculate Monthly
4. Calculate Daily
5. Calculate Continuously

STEP 3

If it's compounded annually, that means the interest is calculated once a year.
So, our n value is $n = 1$ .

STEP 4

We'll use the compound interest formula: $A = P(1 + \frac{r}{n})^{nt}$ .
Here, $P$ is the principal ( $\$9000$ ), $r$ is the interest rate ( $0.035$ ), $n$ is the number of times compounded per year ( $1$ ), and $t$ is the time in years ( $8$ ).

STEP 5

Let's plug in the values: $A = 9000(1 + \frac{0.035}{1})^{1 \cdot 8}$ .

STEP 6

Simplifying, we get $A = 9000(1.035)^{8}$ .

STEP 7

Calculating this gives us $A \approx 11793$ .
So, with annual compounding, we'll have approximately $\$11793$ .

STEP 8

Quarterly means 4 times a year, so $n = 4$ .

STEP 9

Using the same formula, $A = 9000(1 + \frac{0.035}{4})^{4 \cdot 8}$ .

STEP 10

This simplifies to $A = 9000(1.00875)^{32}$ .

STEP 11

Calculating gives us $A \approx 11862$ .
That's about $\$11862$ with quarterly compounding.

STEP 12

Monthly compounding means $n = 12$ .

STEP 13

Our formula becomes $A = 9000(1 + \frac{0.035}{12})^{12 \cdot 8}$ .

STEP 14

Simplifying, $A = 9000(1 + \frac{0.035}{12})^{96} \approx 9000(1.0029167)^{96}$ .

STEP 15

Calculating gives us $A \approx 11878$ , or about $\$11878$ .

STEP 16

For daily compounding, $n = 365$ .

STEP 17

The formula is $A = 9000(1 + \frac{0.035}{365})^{365 \cdot 8}$ .

STEP 18

This simplifies to $A = 9000(1 + \frac{0.035}{365})^{2920} \approx 9000(1.00009589)^{2920}$ .

STEP 19

Calculating gives us $A \approx 11886$ , which is about $\$11886$ .

STEP 20

For continuous compounding, we use a different formula: $A = Pe^{rt}$ .

STEP 21

Plugging in our values, we get $A = 9000e^{0.035 \cdot 8}$ .

STEP 22

This simplifies to $A = 9000e^{0.28}$ .

STEP 23

Calculating this gives us $A \approx 11886$ , or about $\$11886$ .

STEP 24

Here's the completed table:
| Compounding Option | $n$ Value | Result | |---|---|---| | Annually | $n = 1$ | $\$11793$ | | Quarterly | $n = 4$ | $\$11862$ | | Monthly | $n = 12$ | $\$11878$ | | Daily | $n = 365$ | $\$11886$ | | Continuously | Not Applicable | $\$11886$ |

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Studdy Solution

STEP 1

STEP 2

1. Calculate Annually2. Calculate Quarterly3. Calculate Monthly4. Calculate Daily5. Calculate Continuously

STEP 3

If it's compounded **annually**, that means the interest is calculated once a year.So, our **n value** is n=1n = 1n=1.

STEP 4

We'll use the **compound interest formula**: A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}A=P(1+nr​)nt.Here, PPP is the **principal** ($9000\$9000$9000), rrr is the **interest rate** (0.0350.0350.035), nnn is the **number of times compounded per year** (111), and ttt is the **time in years** (888).

STEP 5

Let's plug in the values: A=9000(1+0.0351)1⋅8A = 9000(1 + \frac{0.035}{1})^{1 \cdot 8}A=9000(1+10.035​)1⋅8.

STEP 6

Simplifying, we get A=9000(1.035)8A = 9000(1.035)^{8}A=9000(1.035)8.

STEP 7

Calculating this gives us A≈11793A \approx 11793A≈11793.So, with annual compounding, we'll have approximately $11793\$11793$11793.

STEP 8

**Quarterly** means 4 times a year, so n=4n = 4n=4.

STEP 9

Using the same formula, A=9000(1+0.0354)4⋅8A = 9000(1 + \frac{0.035}{4})^{4 \cdot 8}A=9000(1+40.035​)4⋅8.

STEP 10

This simplifies to A=9000(1.00875)32A = 9000(1.00875)^{32}A=9000(1.00875)32.

STEP 11

Calculating gives us A≈11862A \approx 11862A≈11862.That's about $11862\$11862$11862 with quarterly compounding.

STEP 12

**Monthly** compounding means n=12n = 12n=12.

STEP 13

Our formula becomes A=9000(1+0.03512)12⋅8A = 9000(1 + \frac{0.035}{12})^{12 \cdot 8}A=9000(1+120.035​)12⋅8.

STEP 14

Simplifying, A=9000(1+0.03512)96≈9000(1.0029167)96A = 9000(1 + \frac{0.035}{12})^{96} \approx 9000(1.0029167)^{96}A=9000(1+120.035​)96≈9000(1.0029167)96.

STEP 15

Calculating gives us A≈11878A \approx 11878A≈11878, or about $11878\$11878$11878.

STEP 16

For **daily** compounding, n=365n = 365n=365.

STEP 17

The formula is A=9000(1+0.035365)365⋅8A = 9000(1 + \frac{0.035}{365})^{365 \cdot 8}A=9000(1+3650.035​)365⋅8.

STEP 18

This simplifies to A=9000(1+0.035365)2920≈9000(1.00009589)2920A = 9000(1 + \frac{0.035}{365})^{2920} \approx 9000(1.00009589)^{2920}A=9000(1+3650.035​)2920≈9000(1.00009589)2920.

STEP 19

Calculating gives us A≈11886A \approx 11886A≈11886, which is about $11886\$11886$11886.

STEP 20

For **continuous** compounding, we use a different formula: A=PertA = Pe^{rt}A=Pert.

STEP 21

Plugging in our values, we get A=9000e0.035⋅8A = 9000e^{0.035 \cdot 8}A=9000e0.035⋅8.

STEP 22

This simplifies to A=9000e0.28A = 9000e^{0.28}A=9000e0.28.

STEP 23

Calculating this gives us A≈11886A \approx 11886A≈11886, or about $11886\$11886$11886.

STEP 24

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1. Calculate Annually
2. Calculate Quarterly
3. Calculate Monthly
4. Calculate Daily
5. Calculate Continuously

If it's compounded annually, that means the interest is calculated once a year.
So, our n value is $n = 1$ .

We'll use the compound interest formula: $A = P(1 + \frac{r}{n})^{nt}$ .
Here, $P$ is the principal ( $\$9000$ ), $r$ is the interest rate ( $0.035$ ), $n$ is the number of times compounded per year ( $1$ ), and $t$ is the time in years ( $8$ ).

Let's plug in the values: $A = 9000(1 + \frac{0.035}{1})^{1 \cdot 8}$ .

Simplifying, we get $A = 9000(1.035)^{8}$ .

Calculating this gives us $A \approx 11793$ .
So, with annual compounding, we'll have approximately $\$11793$ .

Quarterly means 4 times a year, so $n = 4$ .

Using the same formula, $A = 9000(1 + \frac{0.035}{4})^{4 \cdot 8}$ .

This simplifies to $A = 9000(1.00875)^{32}$ .

Calculating gives us $A \approx 11862$ .
That's about $\$11862$ with quarterly compounding.

Monthly compounding means $n = 12$ .

Our formula becomes $A = 9000(1 + \frac{0.035}{12})^{12 \cdot 8}$ .

Simplifying, $A = 9000(1 + \frac{0.035}{12})^{96} \approx 9000(1.0029167)^{96}$ .

Calculating gives us $A \approx 11878$ , or about $\$11878$ .

For daily compounding, $n = 365$ .

The formula is $A = 9000(1 + \frac{0.035}{365})^{365 \cdot 8}$ .

This simplifies to $A = 9000(1 + \frac{0.035}{365})^{2920} \approx 9000(1.00009589)^{2920}$ .

Calculating gives us $A \approx 11886$ , which is about $\$11886$ .

For continuous compounding, we use a different formula: $A = Pe^{rt}$ .

Plugging in our values, we get $A = 9000e^{0.035 \cdot 8}$ .

This simplifies to $A = 9000e^{0.28}$ .

Calculating this gives us $A \approx 11886$ , or about $\$11886$ .