There are two maize farms in the countryside. Farm one produces maize at a rate of (\frac{5}{t}) tons per year, where (t) is the number of years. Farm two produces maize at a rate of (\frac{-1.5}{\cos t}) tons per year, where (t) is the number of years. What is the first time that the farms have the same production rate, round your answer to the nearest hundredth.

To find the first time that the two maize farms have the same production rate, we need to solve the equation where the production rates of both farms are equal.

Given:
- Farm 1 produces maize at a rate of \(\frac{5}{t}\) tons per year.
- Farm 2 produces maize at a rate of \(\frac{-1.5}{\cos(t)}\) tons per year.

We need to solve the equation:

\[
\frac{5}{t} = \frac{-1.5}{\cos(t)}
\]

### Step 1: Eliminate the fractions
Multiply both sides of the equation by \( t \cdot \cos(t) \) to eliminate the denominators:

\[
5 \cos(t) = -1.5 t
\]

### Step 2: Rearrange the equation
Rearrange the equation to make it more standard for solving:

\[
5 \cos(t) + 1.5 t = 0
\]

### Step 3: Solve numerically
Since this is a transcendental equation (involving both a trigonometric function and a polynomial term), it cannot be solved algebraically. We need to solve it numerically to find the first time \( t \) when the production rates are equal.

Let me compute this numerically.The first time that the two farms have the same production rate is at \( t = 2.36 \) years, rounded to the nearest hundredth.

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What would happen if the production rates of the farms changed slightly?
2. How does the cosine function influence the behavior of Farm 2's production rate?
3. Can this method be applied if one of the rates were exponential instead of trigonometric?
4. What are other methods for solving transcendental equations besides numerical approaches?
5. How would this solution change if we looked for a later time when the rates are equal?

**Tip:** In equations involving both trigonometric and polynomial functions, numerical solutions (like those from `fsolve`) are often the best approach.

There are two maize farms in the countryside. Farm one produces maize at a rate of (\frac{5}{t}) tons per year, where (t) is the number of years. Farm two produces maize at a rate of (\frac{-1.5}{\cos t}) tons per year, where (t) is the number of years. What is the first time that the farms have the same production rate, round your answer to the nearest hundredth?

Learn how to solve a transcendental equation involving the production rates of two maize farms. This detailed solution covers numerical methods and trigonometric concepts. Suitable for advanced high school or college-level students.

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