Suppose that the machinery in question costs $104000 and earns profit at a continuous rate of $69000 per year. Use an interest rate of 9% per year, compounded continuously. When is the present value of the profit equal to the cost of the machinery? Round your answer to the nearest tenth of a year.

Question

Suppose that the machinery in question costs $104000 and earns profit at a continuous rate of $69000 per year. Use an interest rate of 9% per year, compounded continuously. When is the present value of the profit equal to the cost of the machinery? Round your answer to the nearest tenth of a year.

Answer:

Answer 1

u suck mathmate

Answer 2

To find out when the present value of the profit is equal to the cost of the machinery, we can use the formula for the present value of continuous income:

PV = A / e^(rt)

Where:
PV is the present value
A is the future value (profit)
r is the interest rate per year (as a decimal)
t is the time in years
e is Euler's number (approximately 2.71828)

In this case, the cost of the machinery is $104,000 and the profit per year is $69,000. The interest rate is 9%, which can be expressed as 0.09 in decimal form.

Let's plug in the values and solve for t:

PV = $104,000
A = $69,000
r = 0.09
e = 2.71828

104,000 = 69,000 / e^(0.09t)

To isolate 't', we can multiply both sides of the equation by e^(0.09t):

e^(0.09t) * 104,000 = 69,000

Next, divide both sides by 104,000:

e^(0.09t) = 0.66346

To solve for 't', we will take the natural logarithm of both sides of the equation:

ln(e^(0.09t)) = ln(0.66346)

Using the property of logarithms (ln(e^x) = x):

0.09t = ln(0.66346)

Now, divide both sides by 0.09:

t = ln(0.66346) / 0.09

Using a calculator, the value of t comes out to approximately 4.3 years.

Therefore, the present value of the profit is equal to the cost of the machinery in about 4.3 years. Rounded to the nearest tenth of a year, the answer is 4.3 years.

Answer 3

To find the time when the present value of the profit is equal to the cost of the machinery, we need to calculate the present value of the profit and compare it to the cost of the machinery.

First, let's calculate the present value of the profit using the continuous compound interest formula:

PV = A / e^(rt)

Where:
PV = Present value of the profit
A = Annual profit ($69,000)
r = Interest rate (0.09)
t = Time in years

PV = $69,000 / e^(0.09t)

Now, we need to find the time when the present value of the profit is equal to the cost of the machinery, which is $104,000.

$104,000 = $69,000 / e^(0.09t)

To isolate the exponential term, we can multiply both sides of the equation by e^(0.09t):

$104,000 * e^(0.09t) = $69,000

Divide both sides of the equation by $69,000:

e^(0.09t) = $104,000 / $69,000

e^(0.09t) = 1.5072

To solve for t, we can take the natural logarithm (ln) of both sides of the equation:

ln(e^(0.09t)) = ln(1.5072)

0.09t = ln(1.5072)

Now, divide both sides of the equation by 0.09 to solve for t:

t = ln(1.5072) / 0.09

Using a calculator, we find:

t ≈ 8.18 years

Therefore, the present value of the profit is equal to the cost of the machinery after approximately 8.18 years.

Answer 4

Continuous compounding:

future value = present value * e^rt
where t=number of periods, and r=rate

So assuming the cost of the machinery remains constant, then equate future value of the profit to the cost:
104000 = 69000 * e^rt
e^rt = 104000/69000
r=0.09
Solve for t.
hint: take log to the base e and solve.