Solar Panel Production; The manager of a corporation estimates that t months after it first began production, it was manufacturing the company's 140-watt 12-volt nominal solar panels at the rate of R(t)=(4t/1+t^2)+7√t hundred panels per month. What is the manager's estimate of the number of solar panels manufactured by the company during the second year?

To find the manager's estimate of the number of solar panels manufactured during the second year, we need to calculate the total number of panels produced between \( t = 12 \) months (the end of the first year) and \( t = 24 \) months (the end of the second year).

Given the rate of production function \( R(t) \), which represents the rate at which panels are produced per month in hundreds, the total number of panels produced during the second year can be found by integrating \( R(t) \) from \( t = 12 \) to \( t = 24 \).

The rate function is given by:

\[
R(t) = \frac{4t}{1 + t^2} + 7\sqrt{t}
\]

We need to compute the following definite integral:

\[
\text{Total panels} = \int_{12}^{24} R(t) \, dt
\]

Substituting the given function \( R(t) \):

\[
\text{Total panels} = \int_{12}^{24} \left(\frac{4t}{1 + t^2} + 7\sqrt{t}\right) dt
\]

Let's calculate this integral.The exact value of the definite integral is:

\[
\text{Total panels} = -112\sqrt{3} - 2\log(145) + 2\log(577) + 224\sqrt{6}
\]

This represents the total number of solar panels (in hundreds) produced by the company during the second year. To get the numerical estimate, let's calculate the approximate value of this expression.The manager's estimate of the number of solar panels manufactured during the second year is approximately \( 357.46 \) hundred panels, or about 35,746 panels.

Would you like to see more details on the calculations or have any questions?

Here are some related questions:

1. How does the rate of production change over time?
2. What is the total number of panels produced during the first year?
3. How would the result change if the production rate formula were different?
4. Can the integral be solved using any special techniques?
5. How does the square root term in \( R(t) \) affect the total production?
6. What does the expression \( \frac{4t}{1 + t^2} \) represent in the context of production?
7. What is the significance of integrating the rate function over a specific time interval?
8. How would production be estimated for any given time period?

**Tip:** In problems involving rates, integration is key to determining the total quantity produced or accumulated over time.

Calculate the estimated production of 140-watt 12-volt nominal solar panels during the second year using integration of the production rate function. Learn the step-by-step process and formula application for finding the total number of panels manufactured over time.

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