Math / Algebra

QuestionQuestion 19, 5.3.95 Points: 0 of 1
Use the compound interest formula to determine the final value of the given amount. $\$ 250$ at 5\% compounded daily for 20 years
The final value is $\$ 663.3244263$ . (Round to the nearest cent as needed.)

Studdy Solution

STEP 1

What is this asking? We need to find out how much money we'll have after putting $\$250$ in a savings account for 20 years with a 5% daily interest rate. Watch out! The interest is compounded daily, so we can't just use the simple interest formula.
Also, make sure to round to the nearest cent at the very end!

STEP 2

1. Define the compound interest formula.
2. Plug in the values.
3. Calculate the final amount.

STEP 3

The compound interest formula is given by: $A = P \cdot (1 + \frac{r}{n})^{n \cdot t}$ Where: $A$ is the future value of the investment/loan, including interest. $P$ is the principal investment amount (the initial deposit or loan amount). $r$ is the annual interest rate (decimal). $n$ is the number of times that interest is compounded per year. $t$ is the number of years the money is invested or borrowed for.

STEP 4

Our principal, $P$ , is the initial amount we're investing, which is $\$250$ .

STEP 5

The annual interest rate, $r$ , is 5%, which as a decimal is $0.05$ .

STEP 6

Since the interest is compounded daily, $n$ is 365 (the number of days in a year).

STEP 7

The time, $t$ , is 20 years.

STEP 8

Let's plug these values into our formula: $A = 250 \cdot (1 + \frac{0.05}{365})^{365 \cdot 20}$

STEP 9

First, let's simplify the fraction inside the parentheses: $\frac{0.05}{365} \approx 0.000136987$ So, $A \approx 250 \cdot (1 + 0.000136987)^{365 \cdot 20}$

STEP 10

Now, add one to the result: $1 + 0.000136987 \approx 1.000136987$ So, $A \approx 250 \cdot (1.000136987)^{365 \cdot 20}$

STEP 11

Next, calculate the exponent: $365 \cdot 20 = 7300$ So, $A \approx 250 \cdot (1.000136987)^{7300}$

STEP 12

Now, raise the value inside the parentheses to the power of $7300$ : $(1.000136987)^{7300} \approx 2.73317807$ So, $A \approx 250 \cdot 2.73317807$

STEP 13

Finally, multiply by $250$ to find the final amount: $A \approx 250 \cdot 2.73317807 \approx 683.2945175$

STEP 14

Rounding to the nearest cent, we get $\$683.29$ .

STEP 15

The final value of the investment after 20 years is $\$683.29$ .

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QuestionQuestion 19, 5.3.95 Points: 0 of 1
Use the compound interest formula to determine the final value of the given amount. $\$ 250$ at 5\% compounded daily for 20 years
The final value is $\$ 663.3244263$ . (Round to the nearest cent as needed.)

Studdy Solution

STEP 1

STEP 2

1. Define the compound interest formula.
2. Plug in the values.
3. Calculate the final amount.

STEP 3

STEP 4

Our principal, $P$ , is the initial amount we're investing, which is $\$250$ .

STEP 5

The annual interest rate, $r$ , is 5%, which as a decimal is $0.05$ .

STEP 6

Since the interest is compounded daily, $n$ is 365 (the number of days in a year).

STEP 7

The time, $t$ , is 20 years.

STEP 8

Let's plug these values into our formula: $A = 250 \cdot (1 + \frac{0.05}{365})^{365 \cdot 20}$

STEP 9

First, let's simplify the fraction inside the parentheses: $\frac{0.05}{365} \approx 0.000136987$ So, $A \approx 250 \cdot (1 + 0.000136987)^{365 \cdot 20}$

STEP 10

Now, add one to the result: $1 + 0.000136987 \approx 1.000136987$ So, $A \approx 250 \cdot (1.000136987)^{365 \cdot 20}$

STEP 11

Next, calculate the exponent: $365 \cdot 20 = 7300$ So, $A \approx 250 \cdot (1.000136987)^{7300}$

STEP 12

Now, raise the value inside the parentheses to the power of $7300$ : $(1.000136987)^{7300} \approx 2.73317807$ So, $A \approx 250 \cdot 2.73317807$

STEP 13

Finally, multiply by $250$ to find the final amount: $A \approx 250 \cdot 2.73317807 \approx 683.2945175$

STEP 14

Rounding to the nearest cent, we get $\$683.29$ .

STEP 15

The final value of the investment after 20 years is $\$683.29$ .

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QuestionQuestion 19, 5.3.95 Points: 0 of 1Use the compound interest formula to determine the final value of the given amount. $250\$ 250$250 at 5\% compounded daily for 20 yearsThe final value is $663.3244263\$ 663.3244263$663.3244263. (Round to the nearest cent as needed.)

Studdy Solution

STEP 1

STEP 2

1. Define the compound interest formula.2. Plug in the values.3. Calculate the final amount.

STEP 3

STEP 4

Our **principal**, PPP, is the initial amount we're investing, which is $250\$250$250.

STEP 5

The **annual interest rate**, rrr, is **5%**, which as a decimal is 0.050.050.05.

STEP 6

Since the interest is compounded *daily*, nnn is **365** (the number of days in a year).

STEP 7

The **time**, ttt, is **20 years**.

STEP 8

Let's plug these values into our formula: A=250⋅(1+0.05365)365⋅20A = 250 \cdot (1 + \frac{0.05}{365})^{365 \cdot 20}A=250⋅(1+3650.05​)365⋅20

STEP 9

First, let's simplify the fraction inside the parentheses: 0.05365≈0.000136987\frac{0.05}{365} \approx 0.0001369873650.05​≈0.000136987 So, A≈250⋅(1+0.000136987)365⋅20A \approx 250 \cdot (1 + 0.000136987)^{365 \cdot 20}A≈250⋅(1+0.000136987)365⋅20

STEP 10

Now, add one to the result: 1+0.000136987≈1.0001369871 + 0.000136987 \approx 1.0001369871+0.000136987≈1.000136987 So, A≈250⋅(1.000136987)365⋅20A \approx 250 \cdot (1.000136987)^{365 \cdot 20}A≈250⋅(1.000136987)365⋅20

STEP 11

Next, calculate the exponent: 365⋅20=7300365 \cdot 20 = 7300365⋅20=7300 So, A≈250⋅(1.000136987)7300A \approx 250 \cdot (1.000136987)^{7300}A≈250⋅(1.000136987)7300

STEP 12

Now, raise the value inside the parentheses to the power of 730073007300: (1.000136987)7300≈2.73317807(1.000136987)^{7300} \approx 2.73317807(1.000136987)7300≈2.73317807 So, A≈250⋅2.73317807A \approx 250 \cdot 2.73317807A≈250⋅2.73317807

STEP 13

Finally, multiply by 250250250 to find the final amount: A≈250⋅2.73317807≈683.2945175A \approx 250 \cdot 2.73317807 \approx 683.2945175A≈250⋅2.73317807≈683.2945175

STEP 14

Rounding to the nearest cent, we get $683.29\$683.29$683.29.

STEP 15

The final value of the investment after 20 years is $683.29\$683.29$683.29.

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QuestionQuestion 19, 5.3.95 Points: 0 of 1
Use the compound interest formula to determine the final value of the given amount. $\$ 250$ at 5\% compounded daily for 20 years
The final value is $\$ 663.3244263$ . (Round to the nearest cent as needed.)

1. Define the compound interest formula.
2. Plug in the values.
3. Calculate the final amount.

Our principal, $P$ , is the initial amount we're investing, which is $\$250$ .

The annual interest rate, $r$ , is 5%, which as a decimal is $0.05$ .

Since the interest is compounded daily, $n$ is 365 (the number of days in a year).

The time, $t$ , is 20 years.

Let's plug these values into our formula: $A = 250 \cdot (1 + \frac{0.05}{365})^{365 \cdot 20}$

First, let's simplify the fraction inside the parentheses: $\frac{0.05}{365} \approx 0.000136987$ So, $A \approx 250 \cdot (1 + 0.000136987)^{365 \cdot 20}$

Now, add one to the result: $1 + 0.000136987 \approx 1.000136987$ So, $A \approx 250 \cdot (1.000136987)^{365 \cdot 20}$

Next, calculate the exponent: $365 \cdot 20 = 7300$ So, $A \approx 250 \cdot (1.000136987)^{7300}$

Now, raise the value inside the parentheses to the power of $7300$ : $(1.000136987)^{7300} \approx 2.73317807$ So, $A \approx 250 \cdot 2.73317807$

Finally, multiply by $250$ to find the final amount: $A \approx 250 \cdot 2.73317807 \approx 683.2945175$

Rounding to the nearest cent, we get $\$683.29$ .

The final value of the investment after 20 years is $\$683.29$ .