Problem Analysis #8

To solve the problem, we need to find the volume $V$ of the solid obtained by rotating triangle $ABD$ about the $z$-axis and then determine $p + q$ from the expression $V = \frac{p\pi}{q}$.

Step 1: Determine the coordinates of point $D$.

Point $D$ is the midpoint of $AC$. The coordinates of $A$ and $C$ are $(1, 0, 0)$ and $(0, 0, 1)$, respectively. The midpoint $D$ is calculated as:

Step 2: Find the equations of the lines $AB$, $AD$, and $BD$.

• Line $AB$: Connects $A(1, 0, 0)$ and $B(0, 1, 0)$. The parametric equations are:

  $$x = 1 - t, \quad y = t, \quad z = 0 \quad \text{for} \quad 0 \leq t \leq 1.$$

• Line $AD$: Connects $A(1, 0, 0)$ and $D\left( \frac{1}{2}, 0, \frac{1}{2} \right)$. The parametric equations are:

  $$x = 1 - \frac{1}{2}s, \quad y = 0, \quad z = \frac{1}{2}s \quad \text{for} \quad 0 \leq s \leq 1.$$

• Line $BD$: Connects $B(0, 1, 0)$ and $D\left( \frac{1}{2}, 0, \frac{1}{2} \right)$. The parametric equations are:

  $$x = \frac{1}{2}r, \quad y = 1 - r, \quad z = \frac{1}{2}r \quad \text{for} \quad 0 \leq r \leq 1.$$

Step 3: Calculate the volume using the method of disks/washers.

The volume $V$ is obtained by integrating the area of the disks formed by rotating the triangle about the $z$-axis. The radius of each disk is the distance from the $z$-axis to the point on the triangle.

For the region bounded by $AB$ and $AD$, the radius $r$ is $\sqrt{x^2 + y^2}$. Substituting the parametric equations of $AB$ and $AD$, we get:

The volume integral is:

Evaluating the integral:

Thus, the volume $V$ is:

Step 4: Identify $p$ and $q$ and find $p + q$.

From $V = \frac{2\pi}{3}$, we have $p = 2$ and $q = 3$. Therefore:

The final answer is: