Area bounded by the curves satisfying the conditions `(x^(2))/(25)+(y^(2))/(36)le 1le(x)/(5)+(y)/(6)` is given by

Options

A. `15((pi)/(2)-2)" sq. units"`
B. `(15)/(4)((pi)/(2)-1)" sq. units"`
C. `30(pi-1)" sq. units"`
D. `(15)/(2)(pi-2)" sq. units"`

Answer

To find the area bounded by the curves given by the conditions \(\frac{x^2}{25} + \frac{y^2}{36} \leq 1\) and \(\frac{x}{5} + \frac{y}{6} \geq 1\), we can follow these steps:

### Step 1: Identify the curves
The first inequality \(\frac{x^2}{25} + \frac{y^2}{36} \leq 1\) represents an ellipse centered at the origin with semi-major axis \(b = 6\) (along the y-axis) and semi-minor axis \(a = 5\) (along the x-axis).

The second inequality \(\frac{x}{5} + \frac{y}{6} \geq 1\) represents a line with x-intercept \(5\) and y-intercept \(6\).

### Step 2: Sketch the curves
1. **Ellipse**: The ellipse can be sketched by plotting points where it intersects the axes:
- At \(x = 0\), \(y = 6\) (point (0, 6))
- At \(y = 0\), \(x = 5\) (point (5, 0))

The ellipse will be vertically elongated, opening along the y-axis.

2. **Line**: The line can be drawn by connecting the intercepts:
- Point (5, 0) on the x-axis
- Point (0, 6) on the y-axis

### Step 3: Determine the region of interest
To find the area between the two curves, we need to determine the region where the ellipse is inside the line. We can check a point, such as the origin (0, 0):
- For the ellipse: \(\frac{0^2}{25} + \frac{0^2}{36} = 0 \leq 1\) (inside the ellipse)
- For the line: \(\frac{0}{5} + \frac{0}{6} = 0 \geq 1\) (not inside the line)

Thus, the area we are interested in is outside the ellipse and above the line.

### Step 4: Find the area of the ellipse
The area \(A\) of the ellipse is given by the formula:
\[ A = \pi \cdot a \cdot b \]
where \(a = 5\) and \(b = 6\):
\[ A = \pi \cdot 5 \cdot 6 = 30\pi \]

### Step 5: Calculate the area of the triangle formed by the line
The triangle formed by the line and the axes has a base of \(5\) and a height of \(6\):
\[ \text{Area of triangle} = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 5 \cdot 6 = 15 \]

### Step 6: Calculate the area of the region of interest
The area bounded by the curves is the area of the triangle minus the area of the ellipse in the first quadrant:
\[ \text{Area} = \text{Area of ellipse} - \text{Area of triangle} \]
Since the ellipse is symmetric, we only consider one-fourth of its area:
\[ \text{Area of one-fourth ellipse} = \frac{1}{4} \cdot 30\pi = 7.5\pi \]
Thus, the area we want is:
\[ \text{Area} = 15 - 7.5\pi \]

### Final Step: Write the final answer
The area bounded by the curves is:
\[ \text{Area} = 15\left(\frac{\pi - 2}{2}\right) \text{ square units} \]