The plane Whose equation is 5x+6y+10z=30 forms a pyramid in the first octant with the coordinate planes. Its volume is

Options

A. 15
B. 21
C. 30
D. 36

Explanation

To find the volume of the pyramid formed by the plane \(5x + 6y + 10z = 30\) in the first octant with the coordinate planes, we will follow these steps:

### Step 1: Find the intercepts of the plane with the axes.

1. **X-intercept**: Set \(y = 0\) and \(z = 0\) in the equation of the plane.
\[ 5x + 6(0) + 10(0) = 30 \implies 5x = 30 \implies x = 6 \]
So, the x-intercept is \((6, 0, 0)\).

2. **Y-intercept**: Set \(x = 0\) and \(z = 0\) in the equation of the plane.
\[ 5(0) + 6y + 10(0) = 30 \implies 6y = 30 \implies y = 5 \]
So, the y-intercept is \((0, 5, 0)\).

3. **Z-intercept**: Set \(x = 0\) and \(y = 0\) in the equation of the plane.
\[ 5(0) + 6(0) + 10z = 30 \implies 10z = 30 \implies z = 3 \]
So, the z-intercept is \((0, 0, 3)\).

### Step 2: Identify the vertices of the pyramid.

The vertices of the pyramid formed by the plane in the first octant are:
- \(A(6, 0, 0)\) (x-intercept)
- \(B(0, 5, 0)\) (y-intercept)
- \(C(0, 0, 3)\) (z-intercept)
- \(O(0, 0, 0)\) (the origin)

### Step 3: Calculate the volume of the pyramid.

The volume \(V\) of a pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

In this case, the base of the pyramid is the triangle formed by points \(A\), \(B\), and \(O\).

1. **Base Area**: The area of triangle \(OAB\) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base \(OA = 6\) and the height \(OB = 5\).
\[ \text{Area} = \frac{1}{2} \times 6 \times 5 = 15 \]

2. **Height**: The height of the pyramid is the z-coordinate of point \(C\), which is \(3\).

Now, substituting the base area and height into the volume formula:
\[ V = \frac{1}{3} \times 15 \times 3 = 15 \]

### Final Answer:
The volume of the pyramid is \(15\) cubic units.

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