Answer :
To find the outer and inner radii of a hollow right circular cylinder, here are the steps taken:
1. Understand the Problem:
- The cylinder has two radii: an outer radius and an inner radius.
- The difference between the outer radius and the inner radius is 1 cm.
- The length (or height) of the cylinder is 14 cm.
- The volume of the metal used in making the cylinder is [tex]\(176 \, \text{cm}^3\)[/tex].
2. Define the Variables:
- Let [tex]\( r \)[/tex] be the inner radius of the cylinder.
- Consequently, the outer radius will be [tex]\( r + 1 \)[/tex] (since the difference between them is 1 cm).
3. Volume of the Hollow Cylinder:
- The volume of a hollow cylinder is given by the formula:
[tex]\[
V = \pi \times \text{length} \times \left((\text{outer radius})^2 - (\text{inner radius})^2\right)
\][/tex]
- Substituting the known values and expressions:
[tex]\[
176 = \pi \times 14 \times \left((r + 1)^2 - r^2\right)
\][/tex]
4. Simplifying the Equation:
- Expand the expression inside the parentheses:
[tex]\[
(r + 1)^2 = r^2 + 2r + 1
\][/tex]
- Subtract [tex]\( r^2 \)[/tex] from it:
[tex]\[
(r + 1)^2 - r^2 = r^2 + 2r + 1 - r^2 = 2r + 1
\][/tex]
- Substitute back into the volume equation:
[tex]\[
176 = \pi \times 14 \times (2r + 1)
\][/tex]
- Simplify:
[tex]\[
176 = 28\pi \times r + 14\pi
\][/tex]
5. Solve for the Inner Radius [tex]\( r \)[/tex]:
- Rearrange the equation to solve for [tex]\( r \)[/tex]:
[tex]\[
176 = 28\pi \times r + 14\pi
\][/tex]
[tex]\[
176 = 14\pi(2r + 1)
\][/tex]
[tex]\[
176 = 28\pi \times r + 14\pi
\][/tex]
- Solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{88 - 7\pi}{14\pi}
\][/tex]
6. Find the Outer Radius:
- The outer radius is [tex]\( r + 1 \)[/tex]:
[tex]\[
\text{Outer radius} = \frac{88 - 7\pi}{14\pi} + 1
\][/tex]
The inner and outer radii of the cylinder are [tex]\(\frac{88 - 7\pi}{14\pi}\)[/tex] cm and [tex]\(1 + \frac{88 - 7\pi}{14\pi}\)[/tex] cm, respectively.
1. Understand the Problem:
- The cylinder has two radii: an outer radius and an inner radius.
- The difference between the outer radius and the inner radius is 1 cm.
- The length (or height) of the cylinder is 14 cm.
- The volume of the metal used in making the cylinder is [tex]\(176 \, \text{cm}^3\)[/tex].
2. Define the Variables:
- Let [tex]\( r \)[/tex] be the inner radius of the cylinder.
- Consequently, the outer radius will be [tex]\( r + 1 \)[/tex] (since the difference between them is 1 cm).
3. Volume of the Hollow Cylinder:
- The volume of a hollow cylinder is given by the formula:
[tex]\[
V = \pi \times \text{length} \times \left((\text{outer radius})^2 - (\text{inner radius})^2\right)
\][/tex]
- Substituting the known values and expressions:
[tex]\[
176 = \pi \times 14 \times \left((r + 1)^2 - r^2\right)
\][/tex]
4. Simplifying the Equation:
- Expand the expression inside the parentheses:
[tex]\[
(r + 1)^2 = r^2 + 2r + 1
\][/tex]
- Subtract [tex]\( r^2 \)[/tex] from it:
[tex]\[
(r + 1)^2 - r^2 = r^2 + 2r + 1 - r^2 = 2r + 1
\][/tex]
- Substitute back into the volume equation:
[tex]\[
176 = \pi \times 14 \times (2r + 1)
\][/tex]
- Simplify:
[tex]\[
176 = 28\pi \times r + 14\pi
\][/tex]
5. Solve for the Inner Radius [tex]\( r \)[/tex]:
- Rearrange the equation to solve for [tex]\( r \)[/tex]:
[tex]\[
176 = 28\pi \times r + 14\pi
\][/tex]
[tex]\[
176 = 14\pi(2r + 1)
\][/tex]
[tex]\[
176 = 28\pi \times r + 14\pi
\][/tex]
- Solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{88 - 7\pi}{14\pi}
\][/tex]
6. Find the Outer Radius:
- The outer radius is [tex]\( r + 1 \)[/tex]:
[tex]\[
\text{Outer radius} = \frac{88 - 7\pi}{14\pi} + 1
\][/tex]
The inner and outer radii of the cylinder are [tex]\(\frac{88 - 7\pi}{14\pi}\)[/tex] cm and [tex]\(1 + \frac{88 - 7\pi}{14\pi}\)[/tex] cm, respectively.
