Answer :
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.
[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.
