We appreciate your visit to The difference between the outer and inner radii of a hollow right circular cylinder of length 14 cm is 1 cm If the volume of. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To solve this problem, we need to find the outer and inner radii of a hollow right circular cylinder, given that:
1. The difference between the outer and inner radii is 1 cm.
2. The volume of the metal used to make the cylinder is 176 cm³.
3. The length (height) of the cylinder is 14 cm.
Let's solve this step by step:
### Step 1: Set up the relationship between the outer and inner radii.
Let [tex]\( r_{\text{outer}} \)[/tex] be the outer radius and [tex]\( r_{\text{inner}} \)[/tex] be the inner radius.
According to the problem, the difference between the outer and inner radii is 1 cm:
[tex]\[ r_{\text{outer}} - r_{\text{inner}} = 1 \][/tex]
### Step 2: Use the formula for the volume of a hollow cylinder.
The volume of a hollow cylinder is given by:
[tex]\[ V = \pi \times (r_{\text{outer}}^2 - r_{\text{inner}}^2) \times \text{length} \][/tex]
Given:
- Volume ([tex]\( V \)[/tex]) = 176 cm³
- Length = 14 cm
Substituting these values into the formula:
[tex]\[ 176 = \pi \times (r_{\text{outer}}^2 - r_{\text{inner}}^2) \times 14 \][/tex]
### Step 3: Solve the equations.
Using the equation from Step 1:
[tex]\[ r_{\text{outer}} = r_{\text{inner}} + 1 \][/tex]
Substitute [tex]\( r_{\text{outer}} = r_{\text{inner}} + 1 \)[/tex] into the volume formula:
[tex]\[ 176 = \pi \times ((r_{\text{inner}} + 1)^2 - r_{\text{inner}}^2) \times 14 \][/tex]
Simplify the expression:
[tex]\[ 176 = \pi \times ((r_{\text{inner}}^2 + 2r_{\text{inner}} + 1) - r_{\text{inner}}^2) \times 14 \][/tex]
[tex]\[ 176 = \pi \times (2r_{\text{inner}} + 1) \times 14 \][/tex]
Divide both sides by [tex]\( 14\pi \)[/tex]:
[tex]\[ \frac{176}{14\pi} = 2r_{\text{inner}} + 1 \][/tex]
Simplify to find [tex]\( r_{\text{inner}} \)[/tex]:
[tex]\[ r_{\text{inner}} = \frac{176}{28\pi} - \frac{1}{2} \][/tex]
Calculate the numerical values:
[tex]\[ r_{\text{inner}} \approx 1.5008 \, \text{cm} \][/tex]
Now, use [tex]\( r_{\text{outer}} = r_{\text{inner}} + 1 \)[/tex]:
[tex]\[ r_{\text{outer}} \approx 1.5008 + 1 = 2.5008 \, \text{cm} \][/tex]
### Conclusion:
- Outer radius ([tex]\( r_{\text{outer}} \)[/tex]): Approximately 2.50 cm
- Inner radius ([tex]\( r_{\text{inner}} \)[/tex]): Approximately 1.50 cm
These are the radii for the hollow right circular cylinder based on the problem's conditions.
1. The difference between the outer and inner radii is 1 cm.
2. The volume of the metal used to make the cylinder is 176 cm³.
3. The length (height) of the cylinder is 14 cm.
Let's solve this step by step:
### Step 1: Set up the relationship between the outer and inner radii.
Let [tex]\( r_{\text{outer}} \)[/tex] be the outer radius and [tex]\( r_{\text{inner}} \)[/tex] be the inner radius.
According to the problem, the difference between the outer and inner radii is 1 cm:
[tex]\[ r_{\text{outer}} - r_{\text{inner}} = 1 \][/tex]
### Step 2: Use the formula for the volume of a hollow cylinder.
The volume of a hollow cylinder is given by:
[tex]\[ V = \pi \times (r_{\text{outer}}^2 - r_{\text{inner}}^2) \times \text{length} \][/tex]
Given:
- Volume ([tex]\( V \)[/tex]) = 176 cm³
- Length = 14 cm
Substituting these values into the formula:
[tex]\[ 176 = \pi \times (r_{\text{outer}}^2 - r_{\text{inner}}^2) \times 14 \][/tex]
### Step 3: Solve the equations.
Using the equation from Step 1:
[tex]\[ r_{\text{outer}} = r_{\text{inner}} + 1 \][/tex]
Substitute [tex]\( r_{\text{outer}} = r_{\text{inner}} + 1 \)[/tex] into the volume formula:
[tex]\[ 176 = \pi \times ((r_{\text{inner}} + 1)^2 - r_{\text{inner}}^2) \times 14 \][/tex]
Simplify the expression:
[tex]\[ 176 = \pi \times ((r_{\text{inner}}^2 + 2r_{\text{inner}} + 1) - r_{\text{inner}}^2) \times 14 \][/tex]
[tex]\[ 176 = \pi \times (2r_{\text{inner}} + 1) \times 14 \][/tex]
Divide both sides by [tex]\( 14\pi \)[/tex]:
[tex]\[ \frac{176}{14\pi} = 2r_{\text{inner}} + 1 \][/tex]
Simplify to find [tex]\( r_{\text{inner}} \)[/tex]:
[tex]\[ r_{\text{inner}} = \frac{176}{28\pi} - \frac{1}{2} \][/tex]
Calculate the numerical values:
[tex]\[ r_{\text{inner}} \approx 1.5008 \, \text{cm} \][/tex]
Now, use [tex]\( r_{\text{outer}} = r_{\text{inner}} + 1 \)[/tex]:
[tex]\[ r_{\text{outer}} \approx 1.5008 + 1 = 2.5008 \, \text{cm} \][/tex]
### Conclusion:
- Outer radius ([tex]\( r_{\text{outer}} \)[/tex]): Approximately 2.50 cm
- Inner radius ([tex]\( r_{\text{inner}} \)[/tex]): Approximately 1.50 cm
These are the radii for the hollow right circular cylinder based on the problem's conditions.
Thanks for taking the time to read The difference between the outer and inner radii of a hollow right circular cylinder of length 14 cm is 1 cm If the volume of. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
