First, sketch the region enclosed by the curves f(x) = √(x+90) and g(x)=1/26x + 250/36 on your own piece of paper. Then compute the area of the region.

To compute the area of the region enclosed by the curves \( f(x) = \sqrt{x+90} \) and \( g(x) = \frac{1}{26}x + \frac{250}{36} \), we will follow these steps:

### 1. **Find the Points of Intersection**:
The first step is to determine the points where the two curves intersect. This will give us the limits of integration for calculating the area.

Set the two functions equal to each other:

\[
\sqrt{x+90} = \frac{1}{26}x + \frac{250}{36}
\]

Square both sides to eliminate the square root:

\[
x + 90 = \left( \frac{1}{26}x + \frac{250}{36} \right)^2
\]

This will give us a quadratic equation that we can solve for \( x \). Let's expand and solve this equation:

\[
(x + 90) = \left( \frac{1}{26}x + \frac{250}{36} \right)^2
\]

Solving this equation may involve quite a bit of algebraic work.

### 2. **Set Up the Integral**:
Once we find the points of intersection, say \( x = a \) and \( x = b \), the area between the curves is given by the integral of the difference between the two functions:

\[
A = \int_a^b \left( f(x) - g(x) \right) dx
\]

This simplifies to:

\[
A = \int_a^b \left( \sqrt{x+90} - \left( \frac{1}{26}x + \frac{250}{36} \right) \right) dx
\]

### 3. **Compute the Integral**:
We then compute the above integral to determine the enclosed area.

Let me solve for the points of intersection and the area.The points of intersection between the two curves occur at approximately \( x = -72.83 \) and \( x = 387.72 \). These values will serve as the limits of integration.

### 4. Set Up and Compute the Integral:
Now, we'll set up the integral to compute the area of the region enclosed between the curves from \( x = -72.83 \) to \( x = 387.72 \):

\[
A = \int_{-72.83}^{387.72} \left( \sqrt{x+90} - \left( \frac{1}{26}x + \frac{250}{36} \right) \right) dx
\]

I'll calculate this integral next.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, if you'd like, you can compute the integral manually using the following setup:

\[
A = \int_{-72.83}^{387.72} \left( \sqrt{x+90} - \left( \frac{1}{26}x + \frac{250}{36} \right) \right) dx
\]

This integral will give you the exact area enclosed by the curves.

Let me know if you have any questions about the process or need more details! Here are some related questions to explore further:

1. What techniques can be used to solve integrals involving square roots?
2. How do you find the points of intersection between two curves algebraically?
3. What are some common applications of finding the area between curves in real-life scenarios?
4. How does the symmetry of functions affect the computation of enclosed areas?
5. What is the significance of definite integrals in determining physical quantities like area and volume?

**Tip:** Always verify your points of intersection before setting up integrals to ensure accurate limits.

Learn how to compute the area of the region enclosed by the curves f(x) = √(x+90) and g(x) = 1/26x + 250/36 using definite integrals. This problem covers the intersection of curves and integration techniques, suitable for students in Grades 11-12 or college-level calculus courses.

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