Question

analyzing statements about a right cone
which statements are true? check all that apply.
□ the radius of the cone is 9 units.
□ the height of the cone is 15 units.
□ the height of the cone is 12 units.
□ the volume of the cone is represented by the expression \\(\frac{1}{3}\pi(15)^2(9)\\).
□ the volume of the cone is represented by the expression \\(\frac{1}{3}\pi(9)^2(12)\\).

Explanation:

Step1: Analyze the cone's radius, height

From the diagram, the radius \( r = 9 \) (the horizontal segment with the right angle). The slant height is 15, and using the Pythagorean theorem (\( h=\sqrt{l^{2}-r^{2}} \), where \( l = 15 \), \( r = 9 \)): \( h=\sqrt{15^{2}-9^{2}}=\sqrt{225 - 81}=\sqrt{144}=12 \). So radius is 9, height is 12.

Step2: Analyze volume formula

The volume of a cone is \( V=\frac{1}{3}\pi r^{2}h \). Substituting \( r = 9 \), \( h = 12 \), we get \( V=\frac{1}{3}\pi(9)^{2}(12) \).

Step3: Evaluate each statement

• "The radius of the cone is 9 units." → True (from diagram).
• "The height of the cone is 15 units." → False (height is 12, 15 is slant height).
• "The height of the cone is 12 units." → True (calculated via Pythagoras).
• "The volume of the cone is represented by \( \frac{1}{3}\pi(15)^{2}(9) \)." → False (uses slant height instead of height).
• "The volume of the cone is represented by \( \frac{1}{3}\pi(9)^{2}(12) \)." → True (matches volume formula with \( r = 9 \), \( h = 12 \)).

Answer:

• The radius of the cone is 9 units.
• The height of the cone is 12 units.
• The volume of the cone is represented by the expression \( \frac{1}{3}\pi(9)^{2}(12) \).