Math / Algebra

QuestionExponential and Logarithmic Functions Finding the initial amount in a word problem on continuous compound...
Mr. and Mrs. Scott hope to send their daughter to college in fourteen years. How much money should they invest now at an interest rate of $8.5 \%$ per year, compounded continuously, in order to be able to contribute $\$ 9000$ to her education?
Do not round any intermediate computations, and round your answer to the nearest cent. \\square$

Studdy Solution

STEP 1

What is this asking? How much money do the Scotts need to invest now to have $\$9000$ in 14 years with an interest rate of $8.5\%$ compounded continuously? Watch out! Don't forget to convert the percentage to a decimal!
Also, make sure to use the correct formula for continuous compound interest.

STEP 2

1. Define the formula
2. Plug in the values
3. Solve for the initial investment

STEP 3

Alright, so we're dealing with continuous compound interest, which means we'll use the formula $A = P \cdot e^{rt}$ .
Here, $A$ is the future value (what we want to have in the end), $P$ is the principal or initial investment (what we're trying to find), $r$ is the interest rate (as a decimal!), and $t$ is the time in years.

STEP 4

We know that the Scotts want to have $A = \$9000$ in $t = 14$ years.
The interest rate is $8.5\%$ , which as a decimal is $r = 0.085$ .
Let's plug these values into our formula: $9000 = P \cdot e^{0.085 \cdot 14}$

STEP 5

Now, we want to isolate $\ P$ .
To do this, we can divide both sides of the equation by $e^{0.085 \cdot 14}$ : $P = \frac{9000}{e^{0.085 \cdot 14}}$

STEP 6

Let's simplify the exponent: $0.085 \cdot 14 = 1.19$ .
So, we have: $P = \frac{9000}{e^{1.19}}$

STEP 7

Now, we can calculate $e^{1.19} \approx 3.29457$ .
So, $P = \frac{9000}{3.29457}$

STEP 8

Finally, let's divide to find the initial investment: $P \approx 2731.76$

STEP 9

The Scotts should invest approximately $\$2731.76$ now to reach their goal.

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Studdy Solution

STEP 1

What is this asking? How much money do the Scotts need to invest now to have $\$9000$ in 14 years with an interest rate of $8.5\%$ compounded continuously? Watch out! Don't forget to convert the percentage to a decimal!
Also, make sure to use the correct formula for continuous compound interest.

STEP 2

1. Define the formula
2. Plug in the values
3. Solve for the initial investment

STEP 3

STEP 4

We know that the Scotts want to have $A = \$9000$ in $t = 14$ years.
The interest rate is $8.5\%$ , which as a decimal is $r = 0.085$ .
Let's plug these values into our formula: $9000 = P \cdot e^{0.085 \cdot 14}$

STEP 5

Now, we want to isolate $\ P$ .
To do this, we can divide both sides of the equation by $e^{0.085 \cdot 14}$ : $P = \frac{9000}{e^{0.085 \cdot 14}}$

STEP 6

Let's simplify the exponent: $0.085 \cdot 14 = 1.19$ .
So, we have: $P = \frac{9000}{e^{1.19}}$

STEP 7

Now, we can calculate $e^{1.19} \approx 3.29457$ .
So, $P = \frac{9000}{3.29457}$

STEP 8

Finally, let's divide to find the initial investment: $P \approx 2731.76$

STEP 9

The Scotts should invest approximately $\$2731.76$ now to reach their goal.

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Studdy Solution

STEP 1

STEP 2

1. Define the formula2. Plug in the values3. Solve for the initial investment

STEP 3

STEP 4

We know that the Scotts want to have A=$9000A = \$9000A=$9000 in t=14t = 14t=14 years.The interest rate is 8.5%8.5\%8.5%, which as a decimal is r=0.085r = 0.085r=0.085.Let's **plug these values** into our formula: 9000=P⋅e0.085⋅149000 = P \cdot e^{0.085 \cdot 14}9000=P⋅e0.085⋅14

STEP 5

Now, we want to **isolate** P\ P P.To do this, we can **divide both sides** of the equation by e0.085⋅14e^{0.085 \cdot 14}e0.085⋅14: P=9000e0.085⋅14P = \frac{9000}{e^{0.085 \cdot 14}}P=e0.085⋅149000​

STEP 6

Let's **simplify the exponent**: 0.085⋅14=1.190.085 \cdot 14 = 1.190.085⋅14=1.19.So, we have: P=9000e1.19P = \frac{9000}{e^{1.19}}P=e1.199000​

STEP 7

Now, we can **calculate** e1.19≈3.29457e^{1.19} \approx 3.29457e1.19≈3.29457.So, P=90003.29457P = \frac{9000}{3.29457}P=3.294579000​

STEP 8

Finally, let's **divide** to find the **initial investment**: P≈2731.76P \approx 2731.76P≈2731.76

STEP 9

The Scotts should invest approximately $2731.76\$2731.76$2731.76 now to reach their goal.

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Define the formula
2. Plug in the values
3. Solve for the initial investment

We know that the Scotts want to have $A = \$9000$ in $t = 14$ years.
The interest rate is $8.5\%$ , which as a decimal is $r = 0.085$ .
Let's plug these values into our formula: $9000 = P \cdot e^{0.085 \cdot 14}$

Now, we want to isolate $\ P$ .
To do this, we can divide both sides of the equation by $e^{0.085 \cdot 14}$ : $P = \frac{9000}{e^{0.085 \cdot 14}}$

Let's simplify the exponent: $0.085 \cdot 14 = 1.19$ .
So, we have: $P = \frac{9000}{e^{1.19}}$

Now, we can calculate $e^{1.19} \approx 3.29457$ .
So, $P = \frac{9000}{3.29457}$

Finally, let's divide to find the initial investment: $P \approx 2731.76$

The Scotts should invest approximately $\$2731.76$ now to reach their goal.