A body of revolution is obtained by rotating a rectangular trapezoid about a line perpendicular to the bases and passing through the apex of the obtuse angle. The minor base and height are 4 and 6 respectively. Find the surface area and volume of the solid formed if the major base of the trapezoid is:

a) 8;

b) 12.

To solve this problem, we need to determine the surface area and volume of the solid formed by revolving a trapezoid around a specific axis.

### Step 1: Defining the trapezoid and rotation axis
You have a rectangular trapezoid with the following properties:
- **Minor base** \( b_1 = 4 \)
- **Height** \( h = 6 \)
- Two different **major bases**, depending on the case:
  - (a) \( b_2 = 8 \)
  - (b) \( b_2 = 12 \)

The trapezoid is rotated around a line perpendicular to the bases that passes through the apex of the obtuse angle (where the major base and the height meet).

### Step 2: Setup for Rotation
When the trapezoid is rotated about this axis, we generate a solid of revolution. This shape is a **frustum of a cone** because of the sloping side formed by the difference between the major and minor bases.

#### For both cases:
- The radius of the bottom of the frustum will be the length of the major base \( b_2 \).
- The radius of the top of the frustum will be the length of the minor base \( b_1 = 4 \).
- The height of the frustum is equal to the height of the trapezoid, \( h = 6 \).

### Step 3: Volume of a Frustum of a Cone
The formula for the volume of a frustum of a cone is given by:
\[
V = \frac{1}{3} \pi h (r_1^2 + r_1 r_2 + r_2^2)
\]
where:
- \( r_1 \) is the radius of the smaller base,
- \( r_2 \) is the radius of the larger base,
- \( h \) is the height of the frustum.

### Step 4: Surface Area of a Frustum of a Cone
The formula for the lateral surface area of a frustum of a cone is:
\[
A = \pi (r_1 + r_2) s
\]
where \( s \) is the slant height of the frustum, which can be found using the Pythagorean theorem:
\[
s = \sqrt{(r_2 - r_1)^2 + h^2}
\]

Now, let's calculate these values for both cases.

### Case (a): \( b_2 = 8 \)

For this case:
- \( r_1 = 4 \), \( r_2 = 8 \), and \( h = 6 \).

#### Volume:
\[
V = \frac{1}{3} \pi (6) (4^2 + 4 \cdot 8 + 8^2)
= \frac{1}{3} \pi (6) (16 + 32 + 64)
= \frac{1}{3} \pi (6) \cdot 112
= 224 \pi \, \text{cubic units}
\]

#### Slant height:
\[
s = \sqrt{(8 - 4)^2 + 6^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}
\]

#### Surface Area:
\[
A = \pi (4 + 8) \cdot 2\sqrt{13} = \pi (12) \cdot 2\sqrt{13} = 24\pi\sqrt{13} \, \text{square units}
\]

### Case (b): \( b_2 = 12 \)

For this case:
- \( r_1 = 4 \), \( r_2 = 12 \), and \( h = 6 \).

#### Volume:
\[
V = \frac{1}{3} \pi (6) (4^2 + 4 \cdot 12 + 12^2)
= \frac{1}{3} \pi (6) (16 + 48 + 144)
= \frac{1}{3} \pi (6) \cdot 208
= 416 \pi \, \text{cubic units}
\]

#### Slant height:
\[
s = \sqrt{(12 - 4)^2 + 6^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]

#### Surface Area:
\[
A = \pi (4 + 12) \cdot 10 = \pi (16) \cdot 10 = 160 \pi \, \text{square units}
\]

### Final Results:
- **(a) Major base = 8:**
  - Volume = \( 224 \pi \) cubic units
  - Surface Area = \( 24 \pi \sqrt{13} \) square units
- **(b) Major base = 12:**
  - Volume = \( 416 \pi \) cubic units
  - Surface Area = \( 160 \pi \) square units

Would you like more details, or do you have any questions?

### Follow-up questions:
1. How does the change in the major base affect the volume of the frustum?
2. Why does the surface area formula require the slant height instead of the height?
3. What would the volume be if the height of the trapezoid were doubled?
4. How do the calculations change if we rotate the trapezoid about a different axis?
5. Can this method be generalized for any trapezoid?

### Tip:
For solids of revolution, always ensure you identify the right axis of rotation, as this directly impacts the resulting shape.

A body of revolution is obtained by rotating a rectangular trapezoid about a line perpendicular to the bases and passing through the apex of the obtuse angle. The minor base and height are 4 and 6 respectively. Find the surface area and volume of the solid formed if the major base of the trapezoid is: a) 8; b) 12.

Learn how to calculate the surface area and volume of a solid formed by rotating a trapezoid. The minor base is 4 units, the height is 6 units, and the major base is either 8 or 12 units. This detailed solution uses geometry concepts, including frustum volume and surface area formulas. Suitable for students in Grades 10-12.

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