Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Let's analyze the problem step by step.
We are given:
- The base diameter of the cylinder is [tex]\( x \)[/tex] units.
- The volume of the cylinder is [tex]\( \pi x^3 \)[/tex] cubic units.
From this information, we can derive certain properties of the cylinder.
### Step 1: Determine the radius of the cylinder.
The diameter [tex]\( d \)[/tex] of the cylinder is given as [tex]\( x \)[/tex] units.
The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{x}{2} \][/tex]
### Step 2: Calculate the area of the cylinder's base.
The area [tex]\( A \)[/tex] of a circle (which forms the base of the cylinder) is given by the formula:
[tex]\[ A = \pi r^2 \][/tex]
Substituting [tex]\( r = \frac{x}{2} \)[/tex]:
[tex]\[ A = \pi \left(\frac{x}{2}\right)^2 = \pi \frac{x^2}{4} = \frac{1}{4} \pi x^2 \][/tex]
Therefore, the area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units, which confirms one of the given options.
### Step 3: Relate volume to height.
The volume [tex]\( V \)[/tex] of a cylinder is given by:
[tex]\[ V = \pi r^2 h \][/tex]
We know the volume [tex]\( V = \pi x^3 \)[/tex] and the radius [tex]\( r = \frac{x}{2} \)[/tex]. Substitute these into the volume formula:
[tex]\[ \pi x^3 = \pi \left(\frac{x}{2}\right)^2 h \][/tex]
[tex]\[ \pi x^3 = \pi \left(\frac{x^2}{4}\right) h \][/tex]
[tex]\[ \pi x^3 = \frac{\pi x^2}{4} h \][/tex]
Now, solve for [tex]\( h \)[/tex]:
[tex]\[ x^3 = \frac{x^2}{4} h \][/tex]
[tex]\[ 4 x^3 = x^2 h \][/tex]
[tex]\[ h = 4 x \][/tex]
So, the height of the cylinder is [tex]\( 4 x \)[/tex] units, confirming another given option.
### Conclusion:
Based on the analysis:
- The radius of the cylinder is [tex]\( \frac{x}{2} \)[/tex] units, not [tex]\( 2 x \)[/tex] units.
- The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units. (True)
- The area of the cylinder's base is not [tex]\( \frac{1}{2} \pi x^2 \)[/tex] square units.
- The height of the cylinder is not [tex]\( 2 x \)[/tex] units.
- The height of the cylinder is [tex]\( 4 x \)[/tex] units. (True)
Therefore, the true statements are:
1. The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units.
2. The height of the cylinder is [tex]\( 4 x \)[/tex] units.
We are given:
- The base diameter of the cylinder is [tex]\( x \)[/tex] units.
- The volume of the cylinder is [tex]\( \pi x^3 \)[/tex] cubic units.
From this information, we can derive certain properties of the cylinder.
### Step 1: Determine the radius of the cylinder.
The diameter [tex]\( d \)[/tex] of the cylinder is given as [tex]\( x \)[/tex] units.
The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{x}{2} \][/tex]
### Step 2: Calculate the area of the cylinder's base.
The area [tex]\( A \)[/tex] of a circle (which forms the base of the cylinder) is given by the formula:
[tex]\[ A = \pi r^2 \][/tex]
Substituting [tex]\( r = \frac{x}{2} \)[/tex]:
[tex]\[ A = \pi \left(\frac{x}{2}\right)^2 = \pi \frac{x^2}{4} = \frac{1}{4} \pi x^2 \][/tex]
Therefore, the area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units, which confirms one of the given options.
### Step 3: Relate volume to height.
The volume [tex]\( V \)[/tex] of a cylinder is given by:
[tex]\[ V = \pi r^2 h \][/tex]
We know the volume [tex]\( V = \pi x^3 \)[/tex] and the radius [tex]\( r = \frac{x}{2} \)[/tex]. Substitute these into the volume formula:
[tex]\[ \pi x^3 = \pi \left(\frac{x}{2}\right)^2 h \][/tex]
[tex]\[ \pi x^3 = \pi \left(\frac{x^2}{4}\right) h \][/tex]
[tex]\[ \pi x^3 = \frac{\pi x^2}{4} h \][/tex]
Now, solve for [tex]\( h \)[/tex]:
[tex]\[ x^3 = \frac{x^2}{4} h \][/tex]
[tex]\[ 4 x^3 = x^2 h \][/tex]
[tex]\[ h = 4 x \][/tex]
So, the height of the cylinder is [tex]\( 4 x \)[/tex] units, confirming another given option.
### Conclusion:
Based on the analysis:
- The radius of the cylinder is [tex]\( \frac{x}{2} \)[/tex] units, not [tex]\( 2 x \)[/tex] units.
- The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units. (True)
- The area of the cylinder's base is not [tex]\( \frac{1}{2} \pi x^2 \)[/tex] square units.
- The height of the cylinder is not [tex]\( 2 x \)[/tex] units.
- The height of the cylinder is [tex]\( 4 x \)[/tex] units. (True)
Therefore, the true statements are:
1. The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units.
2. The height of the cylinder is [tex]\( 4 x \)[/tex] units.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
