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The base diameter and the height of a cone are both equal to [tex]x[/tex] units.

Which expression represents the volume of the cone, in cubic units?

A. [tex]\pi x^2[/tex]

B. [tex]2 \pi x^3[/tex]

C. [tex]\frac{1}{3} \pi x^2[/tex]

D. [tex]\frac{1}{12} \pi x^3[/tex]


Sagot :

To determine the volume of a cone where the base diameter and the height are both [tex]\(x\)[/tex] units, we'll use the formula for the volume of a cone:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]

1. Identify the radius and height:

- Since the diameter of the base is [tex]\(x\)[/tex], the radius [tex]\(r\)[/tex] is half of the diameter:
[tex]\[ r = \frac{x}{2} \][/tex]

- The height [tex]\(h\)[/tex] of the cone is given as [tex]\(x\)[/tex]:
[tex]\[ h = x \][/tex]

2. Substitute the radius and height into the volume formula:
[tex]\[ V = \frac{1}{3} \pi \left(\frac{x}{2}\right)^2 x \][/tex]

3. Simplify the expression inside the parentheses:
[tex]\[ \left(\frac{x}{2}\right)^2 = \frac{x^2}{4} \][/tex]

4. Substitute [tex]\(\frac{x^2}{4}\)[/tex] back into the volume formula:
[tex]\[ V = \frac{1}{3} \pi \left(\frac{x^2}{4}\right) x \][/tex]

5. Multiply the terms together:
[tex]\[ V = \frac{1}{3} \pi \frac{x^2}{4} x \][/tex]
[tex]\[ V = \frac{1}{3} \pi \frac{x^3}{4} \][/tex]
[tex]\[ V = \frac{\pi x^3}{12} \][/tex]

So, the expression that represents the volume of the cone is:
[tex]\[ \boxed{\frac{1}{12} \pi x^3} \][/tex]

This matches the final option given in your list.