Current Attempt in Progress A recently installed machine earns the company revenue at a continuous rate of 60,000t+45,000 dollars per year during the first six months of operation and at the continuous rate of 75,000 dollars per year after the first six months. The cost of the machine is 156,000 the interest rate is 7% per year, comfoounded continuously and t is time in years since the machine was installed. (a) Find the present value of the revenue earned by the machine during the first year of operation. Round your answer to the nearest integer. i (b) Find how long it will take for the machine to pay for itself; that is, how long it will take for the present value of the revenue to equal the cost of the machine? Round your answer to two decimal places. square years

## Step 1: Calculate the revenue for the first six months.<br />### The revenue rate is $R(t) = 60000t + 45000$. We integrate this from $t=0$ to $t=0.5$ (6 months is half a year) to find the total revenue during this period. The integral is $\int_0^{0.5} (60000t + 45000) dt = [30000t^2 + 45000t]_0^{0.5} = 30000(0.5)^2 + 45000(0.5) = 30000(0.25) + 22500 = 7500 + 22500 = 30000$.<br /><br />## Step 2: Calculate the present value of the revenue for the first six months.<br />### The present value is calculated using the formula $PV = \int_0^{0.5} R(t)e^{-rt} dt$, where $r$ is the interest rate. In our case, $r = 0.07$. So, $PV = \int_0^{0.5} (60000t + 45000)e^{-0.07t} dt$. This integral can be solved using integration by parts. The result is approximately $28861.54$.<br /><br />## Step 3: Calculate the revenue for the second six months.<br />### The revenue rate is constant at $75000$ per year. For six months, the revenue is $75000 \times 0.5 = 37500$.<br /><br />## Step 4: Calculate the present value of the revenue for the second six months.<br />### The present value is calculated as $PV = \int_{0.5}^1 75000e^{-0.07t} dt = 75000[-\frac{1}{0.07}e^{-0.07t}]_{0.5}^1 = -\frac{75000}{0.07}(e^{-0.07} - e^{-0.035}) \approx 35364.88$.<br /><br />## Step 5: Calculate the total present value of the revenue for the first year.<br />### The total present value is the sum of the present values for the first and second six months: $28861.54 + 35364.88 = 64226.42$.<br /><br />## Step 6: Determine how long it takes for the machine to pay for itself.<br />### The cost of the machine is $156000$. We need to find $T$ such that the present value of the revenue equals the cost. The present value for the first six months is calculated as in Step 2. The present value for the remaining time is $\int_{0.5}^T 75000e^{-0.07t} dt$. The sum of these two present values should equal $156000$. Solving for $T$ numerically yields approximately $T \approx 2.22$ years.