Math / Calculus

Question7) Suppose that Bert opens a savings account with $\$ 275$ that accrues interest continuously with an interest rate of $4.5 \%$ . How much money will Bert have at the end of 20 years? Round your answer to two decimal places.

Studdy Solution

STEP 1

What is this asking? How much money will Bert have in his savings account after 20 years, if he starts with $\$275$ and earns 4.5% interest compounded continuously? Watch out! Don't forget that the interest is compounded continuously, so we need to use the special formula for that, not the regular compound interest formula!

STEP 2

1. Define the formula
2. Plug in the values
3. Calculate the result

STEP 3

Alright, so we're dealing with continuously compounded interest, which means we'll use the formula: $A = Pe^{rt}$ .
Here, A is the final amount, P is the principal (the initial amount Bert invested), e is the exponential constant (approximately 2.71828), r is the interest rate (as a decimal), and t is the time in years.

STEP 4

Let's plug in what we know!
Our principal, P, is $\$275$ .
The interest rate, r, is 4.5%, which we write as $0.045$ in decimal form.
The time, t, is 20 years.
So, our formula becomes: $A = 275 \cdot e^{0.045 \cdot 20}$ .

STEP 5

First, let's simplify the exponent: $0.045 \cdot 20 = 0.9$ .
So, our formula is now $A = 275 \cdot e^{0.9}$ .

STEP 6

Now, we can use a calculator to find the value of $e^{0.9}$ , which is approximately 2.4596.
So, $A = 275 \cdot 2.4596$ .

STEP 7

Finally, multiply $275$ by $2.4596$ to get $A \approx 676.39$ .

STEP 8

After 20 years, Bert will have approximately $\$676.39$ in his savings account.

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Question7) Suppose that Bert opens a savings account with $\$ 275$ that accrues interest continuously with an interest rate of $4.5 \%$ . How much money will Bert have at the end of 20 years? Round your answer to two decimal places.

Studdy Solution

STEP 1

STEP 2

1. Define the formula
2. Plug in the values
3. Calculate the result

STEP 3

STEP 4

Let's plug in what we know!
Our principal, P, is $\$275$ .
The interest rate, r, is 4.5%, which we write as $0.045$ in decimal form.
The time, t, is 20 years.
So, our formula becomes: $A = 275 \cdot e^{0.045 \cdot 20}$ .

STEP 5

First, let's simplify the exponent: $0.045 \cdot 20 = 0.9$ .
So, our formula is now $A = 275 \cdot e^{0.9}$ .

STEP 6

Now, we can use a calculator to find the value of $e^{0.9}$ , which is approximately 2.4596.
So, $A = 275 \cdot 2.4596$ .

STEP 7

Finally, multiply $275$ by $2.4596$ to get $A \approx 676.39$ .

STEP 8

After 20 years, Bert will have approximately $\$676.39$ in his savings account.

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Question7) Suppose that Bert opens a savings account with $275\$ 275$275 that accrues interest continuously with an interest rate of 4.5%4.5 \%4.5%. How much money will Bert have at the end of 20 years? Round your answer to two decimal places.

Studdy Solution

STEP 1

STEP 2

1. Define the formula2. Plug in the values3. Calculate the result

STEP 3

STEP 4

Let's **plug in** what we know!Our **principal**, *P*, is $275\$275$275.The **interest rate**, *r*, is 4.5%, which we write as 0.0450.0450.045 in decimal form.The **time**, *t*, is **20 years**.So, our formula becomes: A=275⋅e0.045⋅20A = 275 \cdot e^{0.045 \cdot 20}A=275⋅e0.045⋅20.

STEP 5

First, let's simplify the exponent: 0.045⋅20=0.90.045 \cdot 20 = 0.90.045⋅20=0.9.So, our formula is now A=275⋅e0.9A = 275 \cdot e^{0.9}A=275⋅e0.9.

STEP 6

Now, we can use a calculator to find the value of e0.9e^{0.9}e0.9, which is approximately **2.4596**.So, A=275⋅2.4596A = 275 \cdot 2.4596A=275⋅2.4596.

STEP 7

Finally, **multiply** 275275275 by 2.45962.45962.4596 to get A≈676.39A \approx 676.39A≈676.39.

STEP 8

After 20 years, Bert will have approximately $676.39\$676.39$676.39 in his savings account.

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Question7) Suppose that Bert opens a savings account with $\$ 275$ that accrues interest continuously with an interest rate of $4.5 \%$ . How much money will Bert have at the end of 20 years? Round your answer to two decimal places.

1. Define the formula
2. Plug in the values
3. Calculate the result

Let's plug in what we know!
Our principal, P, is $\$275$ .
The interest rate, r, is 4.5%, which we write as $0.045$ in decimal form.
The time, t, is 20 years.
So, our formula becomes: $A = 275 \cdot e^{0.045 \cdot 20}$ .

First, let's simplify the exponent: $0.045 \cdot 20 = 0.9$ .
So, our formula is now $A = 275 \cdot e^{0.9}$ .

Now, we can use a calculator to find the value of $e^{0.9}$ , which is approximately 2.4596.
So, $A = 275 \cdot 2.4596$ .

Finally, multiply $275$ by $2.4596$ to get $A \approx 676.39$ .

After 20 years, Bert will have approximately $\$676.39$ in his savings account.