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Math  /  Calculus

Question[0/1.21 Points] DETAILS MY NOTES WANEFMAC8 14.4.025.
Find the total value TV of the given income stream and also find its future value FV˙F \dot{V} (at the end of the given interval) using the given interest rate. (Round your answers to the nearest cen R(t)=31,000,0≤t≤12, at 4%R(t)=31,000,0 \leq t \leq 12, \text { at } 4 \% TV=$T V=\$ □\square FV=$F V=\$ □\square

Studdy Solution

STEP 1

1. The income stream is continuous with a rate function R(t)=31,000 R(t) = 31,000 dollars per year.
2. The time interval is from t=0 t = 0 to t=12 t = 12 years.
3. The interest rate is 4% 4\% per annum, compounded continuously.

STEP 2

1. Calculate the total value (TV) of the income stream.
2. Calculate the future value (FV) of the income stream at the end of the interval.

STEP 3

Calculate the total value (TV) of the income stream using the formula for the present value of a continuous income stream:
TV=∫012R(t) dt TV = \int_{0}^{12} R(t) \, dt
Since R(t)=31,000 R(t) = 31,000 , the integral becomes:
TV=∫01231,000 dt TV = \int_{0}^{12} 31,000 \, dt

STEP 4

Evaluate the integral:
TV=31,000×[t]012 TV = 31,000 \times \left[ t \right]_{0}^{12} =31,000×(12−0) = 31,000 \times (12 - 0) =31,000×12 = 31,000 \times 12 =372,000 = 372,000
The total value of the income stream is:
TV=$372,000 TV = \$372,000

STEP 5

Calculate the future value (FV) of the income stream at the end of the interval using the formula for the future value of a continuous income stream:
FV=TV×ert FV = TV \times e^{rt}
where r=0.04 r = 0.04 and t=12 t = 12 .
Substitute the values:
FV=372,000×e0.04×12 FV = 372,000 \times e^{0.04 \times 12}

STEP 6

Calculate the exponential term and the future value:
FV=372,000×e0.48 FV = 372,000 \times e^{0.48}
Using a calculator to find e0.48≈1.617 e^{0.48} \approx 1.617 :
FV=372,000×1.617 FV = 372,000 \times 1.617 ≈601,524 \approx 601,524
Round to the nearest cent:
FV=$601,524.00 FV = \$601,524.00
The total value of the income stream is:
TV=$372,000 TV = \$372,000
The future value of the income stream is:
FV=$601,524.00 FV = \$601,524.00

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