Math / Algebra

QuestionHow much would you need to deposit in an account now in order to have $\$ 6000$ in the account in 10 years? Assume the account earns $6 \%$ interest compounded continously. \\square$

Studdy Solution

STEP 1

What is this asking? How much money do we need to put in an account today to have \$6000 in 10 years, if the account grows continuously at a 6% interest rate? Watch out! Don't mix up the formulas for different types of interest calculations!
Continuous compounding has its own special formula.

STEP 2

1. Set up the continuous compound interest formula.
2. Plug in the given values.
3. Solve for the initial deposit.

STEP 3

Alright, so we're dealing with continuous compound interest, which means our money is growing constantly!
The formula for this magical growth is: $A = Pe^{rt}$ Where: $A$ is the amount of money we'll have in the future, $P$ is the principal (the initial amount we deposit), $r$ is the interest rate (as a decimal), $t$ is the time (in years), and $e$ is the magical mathematical constant approximately equal to 2.718.

STEP 4

We know we want to have $A = \$6000$ in $t = 10$ years, and our interest rate is $r = 0.06$ (6% as a decimal).
Let's plug these values into our formula: $6000 = Pe^{0.06 \cdot 10}$

STEP 5

Now, we just need to solve for $P$ , our initial deposit.
First, let's simplify the exponent: $6000 = Pe^{0.6}$

STEP 6

To isolate $P$ , we'll divide both sides of the equation by $e^{0.6}$ : $\frac{6000}{e^{0.6}} = \frac{Pe^{0.6}}{e^{0.6}}$

STEP 7

This simplifies to: $P = \frac{6000}{e^{0.6}}$

STEP 8

Now, grab your calculator and compute $e^{0.6}$ , which is approximately 1.822.
So, we have: $P = \frac{6000}{1.822}$

STEP 9

Finally, divide 6000 by 1.822 to find the initial deposit: $P \approx 3292.98$

STEP 10

Therefore, we need to deposit approximately $\$3292.98$ today to have $\$6000$ in 10 years.

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QuestionHow much would you need to deposit in an account now in order to have $\$ 6000$ in the account in 10 years? Assume the account earns $6 \%$ interest compounded continously. \\square$

Studdy Solution

STEP 1

What is this asking? How much money do we need to put in an account today to have \$6000 in 10 years, if the account grows continuously at a 6% interest rate? Watch out! Don't mix up the formulas for different types of interest calculations!
Continuous compounding has its own special formula.

STEP 2

1. Set up the continuous compound interest formula.
2. Plug in the given values.
3. Solve for the initial deposit.

STEP 3

STEP 4

We know we want to have $A = \$6000$ in $t = 10$ years, and our interest rate is $r = 0.06$ (6% as a decimal).
Let's plug these values into our formula: $6000 = Pe^{0.06 \cdot 10}$

STEP 5

Now, we just need to solve for $P$ , our initial deposit.
First, let's simplify the exponent: $6000 = Pe^{0.6}$

STEP 6

To isolate $P$ , we'll divide both sides of the equation by $e^{0.6}$ : $\frac{6000}{e^{0.6}} = \frac{Pe^{0.6}}{e^{0.6}}$

STEP 7

This simplifies to: $P = \frac{6000}{e^{0.6}}$

STEP 8

Now, grab your calculator and compute $e^{0.6}$ , which is approximately 1.822.
So, we have: $P = \frac{6000}{1.822}$

STEP 9

Finally, divide 6000 by 1.822 to find the initial deposit: $P \approx 3292.98$

STEP 10

Therefore, we need to deposit approximately $\$3292.98$ today to have $\$6000$ in 10 years.

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QuestionHow much would you need to deposit in an account now in order to have $6000\$ 6000$6000 in the account in 10 years? Assume the account earns 6%6 \%6% interest compounded continously. \ \square$

Studdy Solution

STEP 1

What is this asking? How much money do we need to put in an account *today* to have \$6000 in 10 years, if the account grows continuously at a 6% interest rate? Watch out! Don't mix up the formulas for different types of interest calculations!Continuous compounding has its own special formula.

STEP 2

1. Set up the continuous compound interest formula.2. Plug in the given values.3. Solve for the initial deposit.

STEP 3

STEP 4

We know we want to have A=$6000A = \$6000A=$6000 in t=10t = 10t=10 years, and our interest rate is r=0.06r = 0.06r=0.06 (6% as a decimal).Let's plug these values into our formula: 6000=Pe0.06⋅10 6000 = Pe^{0.06 \cdot 10} 6000=Pe0.06⋅10

STEP 5

Now, we just need to solve for PPP, our **initial deposit**.First, let's simplify the exponent: 6000=Pe0.6 6000 = Pe^{0.6} 6000=Pe0.6

STEP 6

To isolate PPP, we'll divide both sides of the equation by e0.6e^{0.6}e0.6: 6000e0.6=Pe0.6e0.6 \frac{6000}{e^{0.6}} = \frac{Pe^{0.6}}{e^{0.6}} e0.66000​=e0.6Pe0.6​

STEP 7

This simplifies to: P=6000e0.6 P = \frac{6000}{e^{0.6}} P=e0.66000​

STEP 8

Now, grab your calculator and compute e0.6e^{0.6}e0.6, which is approximately **1.822**.So, we have: P=60001.822 P = \frac{6000}{1.822} P=1.8226000​

STEP 9

Finally, divide 6000 by 1.822 to find the **initial deposit**: P≈3292.98 P \approx 3292.98 P≈3292.98

STEP 10

Therefore, we need to deposit approximately $3292.98\$3292.98$3292.98 today to have $6000\$6000$6000 in 10 years.

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Studdy solves anything!

Start learning now

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QuestionHow much would you need to deposit in an account now in order to have $\$ 6000$ in the account in 10 years? Assume the account earns $6 \%$ interest compounded continously. \\square$

What is this asking? How much money do we need to put in an account today to have \$6000 in 10 years, if the account grows continuously at a 6% interest rate? Watch out! Don't mix up the formulas for different types of interest calculations!
Continuous compounding has its own special formula.

1. Set up the continuous compound interest formula.
2. Plug in the given values.
3. Solve for the initial deposit.

We know we want to have $A = \$6000$ in $t = 10$ years, and our interest rate is $r = 0.06$ (6% as a decimal).
Let's plug these values into our formula: $6000 = Pe^{0.06 \cdot 10}$

Now, we just need to solve for $P$ , our initial deposit.
First, let's simplify the exponent: $6000 = Pe^{0.6}$

To isolate $P$ , we'll divide both sides of the equation by $e^{0.6}$ : $\frac{6000}{e^{0.6}} = \frac{Pe^{0.6}}{e^{0.6}}$

This simplifies to: $P = \frac{6000}{e^{0.6}}$

Now, grab your calculator and compute $e^{0.6}$ , which is approximately 1.822.
So, we have: $P = \frac{6000}{1.822}$

Finally, divide 6000 by 1.822 to find the initial deposit: $P \approx 3292.98$

Therefore, we need to deposit approximately $\$3292.98$ today to have $\$6000$ in 10 years.