Math  /  Algebra

QuestionIf $3200\$ 3200 is invested at 5.4%5.4 \% : a. What is the amount in the account after 5 years if it is compounded annually? b. What is the amount in the account after 5 years if it is compounded quarterly? c. What is the amount in the account after 5 years if it is compounded continuously? d. How much interest was earned after 5 years in the account when it is compounded continuously? e. How long would it take for the money to double if it was compounded semiannually?

Studdy Solution

STEP 1

What is this asking? We're figuring out how much money grows in an investment account over time, given different ways the interest is calculated! Watch out! Don't mix up the different compounding frequencies – annual, quarterly, and continuous compounding are all slightly different beasts!
Also, remember to actually answer the question about how much *interest* was earned, not just the final balance.

STEP 2

1. Compound Annually
2. Compound Quarterly
3. Compound Continuously
4. Calculate Continuous Interest
5. Doubling Time

STEP 3

We're **compounding annually**, which means the interest is calculated and added to the principal once a year.
The formula we'll use is A=P(1+r)t A = P(1 + r)^t , where AA is the **final amount**, PP is the **principal** ($3200\$3200), rr is the **annual interest rate** (as a decimal, so 0.0540.054), and tt is the **number of years** (55).

STEP 4

Let's plug in the values: A=3200(1+0.054)5 A = 3200(1 + 0.054)^5 .

STEP 5

Calculating the expression inside the parentheses: 1+0.054=1.0541 + 0.054 = 1.054.
Now we have A=3200(1.054)5 A = 3200(1.054)^5 .

STEP 6

Raising 1.0541.054 to the power of 55 gives us approximately 1.2952561.295256.
So, A=32001.295256 A = 3200 \cdot 1.295256 .

STEP 7

Multiplying this by the principal gives us A4144.82 A \approx 4144.82 .
So, after 5 years, the amount in the account will be approximately $4144.82\$4144.82.

STEP 8

For **quarterly compounding**, the interest is calculated and added four times a year.
We'll use a slightly modified formula: A=P(1+rn)nt A = P(1 + \frac{r}{n})^{nt} , where nn is the **number of times interest is compounded per year** (44 in this case).

STEP 9

Plugging in our values: A=3200(1+0.0544)45 A = 3200(1 + \frac{0.054}{4})^{4 \cdot 5} .

STEP 10

Simplifying the fraction inside the parentheses: 0.0544=0.0135 \frac{0.054}{4} = 0.0135 .
So, A=3200(1+0.0135)20=3200(1.0135)20 A = 3200(1 + 0.0135)^{20} = 3200(1.0135)^{20} .

STEP 11

Calculating (1.0135)20(1.0135)^{20} gives us approximately 1.307991.30799.
Therefore, A=32001.30799 A = 3200 \cdot 1.30799 .

STEP 12

Multiplying gives us A4185.57 A \approx 4185.57 .
The amount after 5 years with quarterly compounding is approximately $4185.57\$4185.57.

STEP 13

**Continuous compounding** uses the formula A=Pert A = Pe^{rt} , where ee is the mathematical constant approximately equal to 2.718282.71828.

STEP 14

Substituting our values: A=3200e0.0545=3200e0.27 A = 3200e^{0.054 \cdot 5} = 3200e^{0.27} .

STEP 15

Calculating e0.27e^{0.27} gives us approximately 1.309961.30996.
So, A=32001.30996 A = 3200 \cdot 1.30996 .

STEP 16

Multiplying gives us A4191.88 A \approx 4191.88 .
With continuous compounding, the amount after 5 years is approximately $4191.88\$4191.88.

STEP 17

To find the **interest earned**, we subtract the **initial principal** from the **final amount** calculated in the continuous compounding step.

STEP 18

Interest earned = $4191.88$3200=$991.88 \$4191.88 - \$3200 = \$991.88 .

STEP 19

For **semiannual compounding**, we use the formula A=P(1+rn)nt A = P(1 + \frac{r}{n})^{nt} again, with n=2n = 2.
We want to find tt when A=2PA = 2P, meaning the investment has doubled.

STEP 20

Setting up the equation: 2P=P(1+0.0542)2t 2P = P(1 + \frac{0.054}{2})^{2t} .

STEP 21

Dividing both sides by PP gives 2=(1+0.027)2t=(1.027)2t 2 = (1 + 0.027)^{2t} = (1.027)^{2t} .

STEP 22

Taking the natural logarithm of both sides: ln(2)=ln((1.027)2t) \ln(2) = \ln((1.027)^{2t}) .
Using the logarithm property, we get ln(2)=2tln(1.027) \ln(2) = 2t \cdot \ln(1.027) .

STEP 23

Solving for tt: t=ln(2)2ln(1.027)0.693120.02660.69310.053213.03 t = \frac{\ln(2)}{2 \cdot \ln(1.027)} \approx \frac{0.6931}{2 \cdot 0.0266} \approx \frac{0.6931}{0.0532} \approx 13.03 .
It will take approximately 13.0313.03 years for the investment to double with semiannual compounding.

STEP 24

a. $4144.82\$4144.82 b. $4185.57\$4185.57 c. $4191.88\$4191.88 d. $991.88\$991.88 e. 13.0313.03 years

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