A right triangle with sides 3 cm, 4 cm, and 5 cm is rotated about the side of 3 cm to form a cone. What is the volume of the cone so formed?

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Accepted Answer

Let's analyze the problem step-by-step:

1. The triangle has sides 3 cm, 4 cm, and 5 cm. Since 3² + 4² = 5², it is a right triangle.
2. The triangle is rotated about the side of 3 cm. This side becomes the height (h) of the cone.
3. The side opposite to 3 cm is 4 cm, which becomes the radius (r) of the base of the cone.
4. The hypotenuse (5 cm) becomes the slant height (l) of the cone, but for volume calculation, we only need r and h.

Volume of a cone = (1/3)πr²h
Here, r = 4 cm, h = 3 cm

So,
Volume = (1/3) × π × (4)² × 3
        = (1/3) × π × 16 × 3
        = (1/3) × π × 48
        = 16π cm³

Wait! Let's double-check the assignment of radius and height:
- When the triangle is rotated about the side of 3 cm, the 3 cm side traces a circle of radius 4 cm (the other leg), so the radius is 4 cm, height is 3 cm.

But let's visualize: If the triangle is rotated about the side of 3 cm, the 3 cm side remains fixed (axis), and the 4 cm side sweeps out the base (radius = 4 cm). So, the above calculation is correct.

But the calculation above gives 16π cm³, which matches option 2, not option 1. Let's check the calculation again:

Volume = (1/3) × π × r² × h
r = 4 cm, h = 3 cm
r² = 16
(1/3) × π × 16 × 3 = (1/3) × π × 48 = 16π cm³

So, the correct answer is 16π cm³. But the answer key says 20π cm³ (option 1). Let's check if the radius and height have been swapped.

If we rotate about the side of 3 cm, the height is 4 cm, and the radius is 3 cm?
No, that's not correct. The axis is 3 cm, so the height is 3 cm, and the radius is 4 cm.

Therefore, the correct answer is 16π cm³, which is option 2.

But the answer key says 20π cm³. Let's check if the question wants us to rotate about the side of 4 cm instead. If rotated about 4 cm:
- Height = 4 cm
- Radius = 3 cm
Volume = (1/3) × π × 9 × 4 = (1/3) × π × 36 = 12π cm³

So, that's not matching either.

Therefore, the correct answer is 16π cm³ (option 2), and the answer key may be incorrect.

Answer

Answer

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