Math / Algebra

QuestionNetflix ENG evalua Evaluation p atlas-query Dashboard ALEKS - Mo Untitled for (370 -- www-awu.aleks.com/alekscgi/X/Isl.exe/10_u-IgNslkr7j8P3jH-lis1WHQv7rHh0j2icTeSIC4QOKrKwOC-WYA3XiX_GKyS_7HdaDkbChlvJtY-wEXTV6q...
Exponential and Logarithmic Functions $0 / 3$ Finding the initial amount in a word problem on continuous compound..."
Susan wants to buy a bond that will mature to $\$ 6500$ in eight years. How much should she pay for the bond now if it earns interest at a rate of $3.5 \%$ per year, compounded continuously? 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 should Susan pay now for a bond that will be worth $\$6500$ in eight years with continuous compounding at $3.5\%$ interest per year? Watch out! Don't round any numbers until the final answer!

STEP 2

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

STEP 3

Alright, let's get this party started!
We're dealing with continuous compounding, so we need the formula for continuous compound interest, which is:
$A = P \cdot e^{r \cdot t}$
where: - $A$ is the final amount (what the bond will be worth in the future), - $P$ is the initial amount (what Susan needs to pay now), - $r$ is the rate of interest (expressed as a decimal), - $t$ is the time in years, - $e$ is the base of the natural logarithm, approximately $2.71828$ .

STEP 4

Let's plug in what we know!
We want the bond to mature to $\$6500$ , so $A = 6500$ .
The interest rate is $3.5\%$ , which as a decimal is $0.035$ , so $r = 0.035$ .
The time is eight years, so $t = 8$ .

STEP 5

Now, we need to solve for $P$ , the initial amount.
Let's rearrange our formula to isolate $P$ :
$P = \frac{A}{e^{r \cdot t}}$

STEP 6

Substitute the values we have into this equation:
$P = \frac{6500}{e^{0.035 \cdot 8}}$

STEP 7

Calculate the exponent first:
$0.035 \cdot 8 = 0.28$

STEP 8

Now, calculate $e^{0.28}$ :
$e^{0.28} \approx 1.32313$

STEP 9

Finally, divide to find $P$ :
$P = \frac{6500}{1.32313} \approx 4913.97$

STEP 10

Susan should pay approximately $\$4913.97$ for the bond now.

Was this helpful?

Studdy Solution

STEP 1

What is this asking? How much should Susan pay now for a bond that will be worth $\$6500$ in eight years with continuous compounding at $3.5\%$ interest per year? Watch out! Don't round any numbers until the final answer!

STEP 2

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

STEP 3

STEP 4

Let's plug in what we know!
We want the bond to mature to $\$6500$ , so $A = 6500$ .
The interest rate is $3.5\%$ , which as a decimal is $0.035$ , so $r = 0.035$ .
The time is eight years, so $t = 8$ .

STEP 5

Now, we need to solve for $P$ , the initial amount.
Let's rearrange our formula to isolate $P$ :
$P = \frac{A}{e^{r \cdot t}}$

STEP 6

Substitute the values we have into this equation:
$P = \frac{6500}{e^{0.035 \cdot 8}}$

STEP 7

Calculate the exponent first:
$0.035 \cdot 8 = 0.28$

STEP 8

Now, calculate $e^{0.28}$ :
$e^{0.28} \approx 1.32313$

STEP 9

Finally, divide to find $P$ :
$P = \frac{6500}{1.32313} \approx 4913.97$

STEP 10

Susan should pay approximately $\$4913.97$ for the bond now.

Get Studdy

Studdy solves anything!

Start learning now

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

Studdy Solution

STEP 1

What is this asking? How much should Susan pay now for a bond that will be worth $6500\$6500$6500 in eight years with continuous compounding at 3.5%3.5\%3.5% interest per year? Watch out! Don't round any numbers until the final answer!

STEP 2

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

STEP 3

STEP 4

Let's plug in what we know!We want the bond to mature to $6500\$6500$6500, so A=6500A = 6500A=6500.The interest rate is 3.5%3.5\%3.5%, which as a decimal is 0.0350.0350.035, so r=0.035r = 0.035r=0.035.The time is eight years, so t=8t = 8t=8.

STEP 5

Now, we need to solve for PPP, the initial amount.Let's rearrange our formula to isolate PPP:P=Aer⋅tP = \frac{A}{e^{r \cdot t}}P=er⋅tA​

STEP 6

Substitute the values we have into this equation:P=6500e0.035⋅8P = \frac{6500}{e^{0.035 \cdot 8}}P=e0.035⋅86500​

STEP 7

Calculate the exponent first:0.035⋅8=0.280.035 \cdot 8 = 0.280.035⋅8=0.28

STEP 8

Now, calculate e0.28e^{0.28}e0.28:e0.28≈1.32313e^{0.28} \approx 1.32313e0.28≈1.32313

STEP 9

Finally, divide to find PPP:P=65001.32313≈4913.97P = \frac{6500}{1.32313} \approx 4913.97P=1.323136500​≈4913.97

STEP 10

Susan should pay approximately $4913.97\$4913.97$4913.97 for the bond now.

Get Studdy

Studdy solves anything!

Start learning now

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

What is this asking? How much should Susan pay now for a bond that will be worth $\$6500$ in eight years with continuous compounding at $3.5\%$ interest per year? Watch out! Don't round any numbers until the final answer!

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

Let's plug in what we know!
We want the bond to mature to $\$6500$ , so $A = 6500$ .
The interest rate is $3.5\%$ , which as a decimal is $0.035$ , so $r = 0.035$ .
The time is eight years, so $t = 8$ .

Now, we need to solve for $P$ , the initial amount.
Let's rearrange our formula to isolate $P$ :
$P = \frac{A}{e^{r \cdot t}}$

Substitute the values we have into this equation:
$P = \frac{6500}{e^{0.035 \cdot 8}}$

Calculate the exponent first:
$0.035 \cdot 8 = 0.28$

Now, calculate $e^{0.28}$ :
$e^{0.28} \approx 1.32313$

Finally, divide to find $P$ :
$P = \frac{6500}{1.32313} \approx 4913.97$

Susan should pay approximately $\$4913.97$ for the bond now.