Solved on Feb 22, 2024

Find the present value of $\$ 100,000$ due in 6 years at a continuous 7% interest rate. The present value is $\$ \square$ .

STEP 1

Assumptions
1. The future value amount is $P = \$100,000$ .
2. The time until the amount is due is $t = 6$ years.
3. The interest rate is $k = 7\%$ .
4. The interest is compounded continuously.

STEP 2

To find the present value $P_{0}$ , we use the formula for continuous compounding:
$P_{0} = P \cdot e^{-kt}$
where $e$ is the base of the natural logarithm, approximately equal to $2.71828$ .

STEP 3

Convert the interest rate from a percentage to a decimal by dividing by 100:
$k = 7\% = \frac{7}{100} = 0.07$

STEP 4

Now, plug in the values for $P$ , $k$ , and $t$ into the formula for $P_{0}$ :
$P_{0} = \$100,000 \cdot e^{-0.07 \cdot 6}$

STEP 5

Calculate the exponent:
$-0.07 \cdot 6 = -0.42$

STEP 6

Calculate $e^{-0.42}$ using a calculator or software that has the capability to compute values of $e$ to the power of a number:
$e^{-0.42} \approx 0.657046$

STEP 7

Now, multiply the future value $P$ by $e^{-0.42}$ to find the present value $P_{0}$ :
$P_{0} = \$100,000 \cdot 0.657046$

STEP 8

Calculate the present value $P_{0}$ :
$P_{0} = \$100,000 \cdot 0.657046 \approx \$65,704.60$
The present value of $\$100,000$ is $\$65,704.60$ . (Rounded to the nearest cent as needed.)

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Find the present value of $\$ 100,000$ due in 6 years at a continuous 7% interest rate. The present value is $\$ \square$ .

STEP 1

Assumptions
1. The future value amount is $P = \$100,000$ .
2. The time until the amount is due is $t = 6$ years.
3. The interest rate is $k = 7\%$ .
4. The interest is compounded continuously.

STEP 2

To find the present value $P_{0}$ , we use the formula for continuous compounding:
$P_{0} = P \cdot e^{-kt}$
where $e$ is the base of the natural logarithm, approximately equal to $2.71828$ .

STEP 3

Convert the interest rate from a percentage to a decimal by dividing by 100:
$k = 7\% = \frac{7}{100} = 0.07$

STEP 4

Now, plug in the values for $P$ , $k$ , and $t$ into the formula for $P_{0}$ :
$P_{0} = \$100,000 \cdot e^{-0.07 \cdot 6}$

STEP 5

Calculate the exponent:
$-0.07 \cdot 6 = -0.42$

STEP 6

Calculate $e^{-0.42}$ using a calculator or software that has the capability to compute values of $e$ to the power of a number:
$e^{-0.42} \approx 0.657046$

STEP 7

Now, multiply the future value $P$ by $e^{-0.42}$ to find the present value $P_{0}$ :
$P_{0} = \$100,000 \cdot 0.657046$

STEP 8

Calculate the present value $P_{0}$ :
$P_{0} = \$100,000 \cdot 0.657046 \approx \$65,704.60$
The present value of $\$100,000$ is $\$65,704.60$ . (Rounded to the nearest cent as needed.)

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Find the present value of $100,000\$ 100,000$100,000 due in 6 years at a continuous 7% interest rate. The present value is $□\$ \square$□.

STEP 1

Assumptions1. The future value amount is P=$100,000P = \$100,000P=$100,000.2. The time until the amount is due is t=6t = 6t=6 years.3. The interest rate is k=7%k = 7\%k=7%.4. The interest is compounded continuously.

STEP 2

To find the present value P0P_{0}P0​, we use the formula for continuous compounding:P0=P⋅e−ktP_{0} = P \cdot e^{-kt}P0​=P⋅e−ktwhere eee is the base of the natural logarithm, approximately equal to 2.718282.718282.71828.

STEP 3

Convert the interest rate from a percentage to a decimal by dividing by 100:k=7%=7100=0.07k = 7\% = \frac{7}{100} = 0.07k=7%=1007​=0.07

STEP 4

Now, plug in the values for PPP, kkk, and ttt into the formula for P0P_{0}P0​:P0=$100,000⋅e−0.07⋅6P_{0} = \$100,000 \cdot e^{-0.07 \cdot 6}P0​=$100,000⋅e−0.07⋅6

STEP 5

Calculate the exponent:−0.07⋅6=−0.42-0.07 \cdot 6 = -0.42−0.07⋅6=−0.42

STEP 6

Calculate e−0.42e^{-0.42}e−0.42 using a calculator or software that has the capability to compute values of eee to the power of a number:e−0.42≈0.657046e^{-0.42} \approx 0.657046e−0.42≈0.657046

STEP 7

Now, multiply the future value PPP by e−0.42e^{-0.42}e−0.42 to find the present value P0P_{0}P0​:P0=$100,000⋅0.657046P_{0} = \$100,000 \cdot 0.657046P0​=$100,000⋅0.657046

STEP 8

Calculate the present value P0P_{0}P0​:P0=$100,000⋅0.657046≈$65,704.60P_{0} = \$100,000 \cdot 0.657046 \approx \$65,704.60P0​=$100,000⋅0.657046≈$65,704.60The present value of $100,000\$100,000$100,000 is $65,704.60\$65,704.60$65,704.60. (Rounded to the nearest cent as needed.)

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Find the present value of $\$ 100,000$ due in 6 years at a continuous 7% interest rate. The present value is $\$ \square$ .

Assumptions
1. The future value amount is $P = \$100,000$ .
2. The time until the amount is due is $t = 6$ years.
3. The interest rate is $k = 7\%$ .
4. The interest is compounded continuously.

To find the present value $P_{0}$ , we use the formula for continuous compounding:
$P_{0} = P \cdot e^{-kt}$
where $e$ is the base of the natural logarithm, approximately equal to $2.71828$ .

Convert the interest rate from a percentage to a decimal by dividing by 100:
$k = 7\% = \frac{7}{100} = 0.07$

Now, plug in the values for $P$ , $k$ , and $t$ into the formula for $P_{0}$ :
$P_{0} = \$100,000 \cdot e^{-0.07 \cdot 6}$

Calculate the exponent:
$-0.07 \cdot 6 = -0.42$

Calculate $e^{-0.42}$ using a calculator or software that has the capability to compute values of $e$ to the power of a number:
$e^{-0.42} \approx 0.657046$

Now, multiply the future value $P$ by $e^{-0.42}$ to find the present value $P_{0}$ :
$P_{0} = \$100,000 \cdot 0.657046$

Calculate the present value $P_{0}$ :
$P_{0} = \$100,000 \cdot 0.657046 \approx \$65,704.60$
The present value of $\$100,000$ is $\$65,704.60$ . (Rounded to the nearest cent as needed.)