A machine that costs $100,000 earns a continuous income stream of $10,000 per year. Approximately how long does it take before the present value of the income generated by the machine equals the purchase price? Use an interest rate of 8.5% per year compounded continuously.
I see the way to do it, I just can't seem to get the correct answer when I try. As much explanation as possible would be greatly appreciated
so, what have you tried?
sites like this one should help.
https://courses.lumenlearning.com/businesscalc1/chapter/reading-continuous-income-stream/
Ah, the magical world of finance! Don't worry, I'm here to help make this more fun for you. Let's crunch those numbers!
To find the time it takes before the present value of income generated by the machine equals the purchase price, we'll need to use the formula for continuous compound interest:
PV = C / e^(rt)
Where:
PV is the present value (purchase price),
C is the continuous income stream ($10,000 per year),
e is the mathematical constant approximately equal to 2.71828,
r is the interest rate (8.5% or 0.085), and
t is the time in years that we want to find.
Now, we want to find t when PV equals $100,000. So, let's set up the equation:
100,000 = 10,000 / e^(0.085t)
To make things simpler, let's divide both sides of the equation by 10,000:
10 = 1 / e^(0.085t)
Now, let's take the natural logarithm (ln) of both sides:
ln(10) = ln(1 / e^(0.085t))
Using the properties of logarithms, we can simplify this further:
ln(10) = -0.085t
Finally, divide both sides by -0.085 to isolate t:
t = ln(10) / -0.085
Now all you need to do is grab your trusty calculator and calculate ln(10) divided by -0.085. But be careful, don't drop any numbers! Good luck, my friend!
To find out how long it takes for the present value of the income generated by the machine to equal the purchase price, we can use the concept of present value.
The present value (PV) of an income stream is calculated using the formula:
PV = A/e^(rt)
Where:
PV = present value
A = annual income
e = the base of natural logarithms (approximately equal to 2.71828)
r = interest rate
t = time in years
In this case, the purchase price (which is the present value we're trying to find) matches the future value of the income stream. Therefore, we can rewrite the formula as:
100,000 = 10,000/e^(0.085t)
Now we can solve for t. First, divide both sides of the equation by 10,000:
10 = 1/e^(0.085t)
Then, take the reciprocal of both sides:
1/10 = e^(0.085t)
To isolate e^(0.085t), take the natural logarithm (ln) of both sides:
ln(1/10) = 0.085t
Now, divide both sides of the equation by 0.085:
t = ln(1/10)/0.085
Using a calculator, you can find:
t ≈ 11.096
So it takes approximately 11.096 years for the present value of the income generated by the machine to equal the purchase price.
To find the approximate length of time it takes for the present value of the income generated by the machine to equal the purchase price, we need to calculate the present value of the income stream.
The present value of a continuous income stream can be calculated using the formula:
PV = I / r
Where:
PV = Present Value
I = Income per year
r = Interest rate (in this case, 8.5% per year)
In this case, I = $10,000 and r = 8.5% = 0.085 (as a decimal). Let's calculate the present value:
PV = $10,000 / 0.085
PV ≈ $117,647.06
Now, we need to find out how long it takes for the present value of the income stream to equal the purchase price of $100,000. For this, we can use the formula for compound interest:
PV = A(1 + r)^t
Where:
A = Initial amount (purchase price)
r = Interest rate (in this case, 8.5% per year)
t = Time (in years)
Let's rearrange the formula and solve for t:
A(1 + r)^t = PV
Substituting the values, we get:
$100,000(1 + 0.085)^t = $117,647.06
Now, we can solve for t. Taking the logarithm of both sides can help simplify the equation:
log($100,000(1 + 0.085)^t) = log($117,647.06)
Using logarithmic properties, we can simplify further:
log($100,000) + log(1 + 0.085)^t = log($117,647.06)
By subtracting log($100,000) from both sides:
log(1 + 0.085)^t = log($117,647.06) - log($100,000)
Using logarithmic properties, we can further simplify the equation:
t * log(1.085) = log($117,647.06) - log($100,000)
Now, we can solve for t by dividing both sides by log(1.085):
t = (log($117,647.06) - log($100,000)) / log(1.085)
Using a scientific calculator or software, we can calculate:
t ≈ 3.51 years
Therefore, it takes approximately 3.51 years for the present value of the income generated by the machine to equal the purchase price, given an interest rate of 8.5% per year compounded continuously.