Question

1. do they have the same volume? reason out why?

Explanation:

Step1: Recall Cone Volume Formula

The volume \( V \) of a cone is given by \( V = \frac{1}{3}\pi r^2 h \), where \( r \) is the radius of the base and \( h \) is the height (perpendicular height from the base to the apex).

Step2: Analyze the Two Cones

For both cones:

• Radius \( r = 3 \) (from the diagram, the base radius is 3 for both).
• Height \( h = 5 \) (the perpendicular height from the base to the apex is 5 for both; in the second cone, the height is the perpendicular distance, which is 5, same as the first cone).

Step3: Calculate Volume for Each Cone

Using the formula \( V = \frac{1}{3}\pi r^2 h \):

• For the first cone: \( V_1 = \frac{1}{3}\pi (3)^2 (5) = \frac{1}{3}\pi \times 9 \times 5 = 15\pi \).
• For the second cone: \( V_2 = \frac{1}{3}\pi (3)^2 (5) = \frac{1}{3}\pi \times 9 \times 5 = 15\pi \).

Since \( V_1 = V_2 \), the volumes are the same because they have the same radius and the same perpendicular height, and the volume of a cone depends only on the radius of the base and the perpendicular height (not the slant height).

Answer:

Yes, they have the same volume. The volume of a cone is given by \( V=\frac{1}{3}\pi r^{2}h \). Both cones have the same radius (\( r = 3 \)) and the same perpendicular height (\( h = 5 \)). Substituting these values into the formula gives the same volume (\( 15\pi \)) for both cones.