The total area (in sq. unit) of the triangles formed by the graph of `4 x + 5 y = 40 , x-` axis , y - axis and x = 5 and y = 4 is

Options

A. 10
B. 20
C. 30
D. 40

Answer

To find the total area of the triangles formed by the graph of \(4x + 5y = 40\), the x-axis, the y-axis, and the lines \(x = 5\) and \(y = 4\), we can follow these steps:

### Step 1: Find the intercepts of the line \(4x + 5y = 40\)
To find the x-intercept, set \(y = 0\):
\[ 4x + 5(0) = 40 \implies 4x = 40 \implies x = 10 \]
So, the x-intercept is \((10, 0)\).

To find the y-intercept, set \(x = 0\):
\[ 4(0) + 5y = 40 \implies 5y = 40 \implies y = 8 \]
So, the y-intercept is \((0, 8)\).

### Step 2: Identify the vertices of the triangle
The vertices of the triangle formed by the x-axis, y-axis, and the line \(4x + 5y = 40\) are:
1. \((10, 0)\) - x-intercept
2. \((0, 8)\) - y-intercept
3. \((0, 0)\) - origin

### Step 3: Calculate the area of the triangle
The area \(A\) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the x-intercept (10) and the height is the y-intercept (8):
\[ A = \frac{1}{2} \times 10 \times 8 = \frac{80}{2} = 40 \text{ sq. units} \]

### Step 4: Consider the triangle formed by \(x = 5\) and \(y = 4\)
Next, we need to find the area of the triangle formed by the points \((5, 0)\), \((0, 4)\), and \((5, 4)\).

### Step 5: Calculate the area of the second triangle
The base of this triangle is the segment on the x-axis from \((0, 0)\) to \((5, 0)\), which has a length of 5. The height is the segment on the y-axis from \((0, 0)\) to \((0, 4)\), which has a length of 4:
\[ A_2 = \frac{1}{2} \times 5 \times 4 = \frac{20}{2} = 10 \text{ sq. units} \]

### Step 6: Calculate the total area
Now, we add the areas of both triangles:
\[ \text{Total Area} = A + A_2 = 40 + 10 = 50 \text{ sq. units} \]

### Final Answer
The total area of the triangles formed is **50 sq. units**.

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