To solve these problems, let's go through each step in detail.
### Problem 1: Depth of the Oil in the Drum
1. Determine the volume of oil and the dimensions of the drum:
- Volume of oil: [tex]\( 99 \, \text{cm}^3 \)[/tex]
- Drum diameter: [tex]\( 60 \, \text{cm} \)[/tex]
2. Calculate the radius of the drum:
- Radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{60 \, \text{cm}}{2} = 30 \, \text{cm}
\][/tex]
3. Using the volume formula for a cylinder to find the depth (or height) [tex]\( h \)[/tex]:
- The volume [tex]\( V \)[/tex] of a cylinder is given by [tex]\( V = \pi r^2 h \)[/tex].
- Rearrange to solve for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{V}{\pi r^2}
\][/tex]
- Plug in the values:
[tex]\[
h = \frac{99 \, \text{cm}^3}{\pi (30 \, \text{cm})^2}
\][/tex]
Simplify the calculation:
[tex]\[
h = \frac{99 \, \text{cm}^3}{\pi \times 900 \, \text{cm}^2}
\][/tex]
[tex]\[
h = \frac{99}{2827.4333882308137} \approx 0.035014 \, \text{cm}
\][/tex]
- Therefore, the depth of the oil in the drum is approximately [tex]\( 0.035 \, \text{cm} \)[/tex].
### Problem 2: Volume of Water in the Pipe
1. Determine the dimensions of the cylindrical pipe:
- Length of the pipe: [tex]\( 1 \, \text{m} = 100 \, \text{cm} \)[/tex]
- Pipe diameter: [tex]\( 7 \, \text{cm} \)[/tex]
2. Calculate the radius of the pipe:
- Radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{7 \, \text{cm}}{2} = 3.5 \, \text{cm}
\][/tex]
3. Using the volume formula for a cylinder to find the volume [tex]\( V \)[/tex]:
- Volume [tex]\( V \)[/tex] of a cylinder is given by [tex]\( V = \pi r^2 h \)[/tex].
- Here, radius [tex]\( r = 3.5 \, \text{cm} \)[/tex] and height [tex]\( h = 100 \, \text{cm} \)[/tex]:
[tex]\[
V = \pi (3.5 \, \text{cm})^2 (100 \, \text{cm})
\][/tex]
Simplify the calculation:
[tex]\[
V = \pi \times 12.25 \, \text{cm}^2 \times 100 \, \text{cm}
\][/tex]
[tex]\[
V \approx 3848.451 \, \text{cm}^3
\][/tex]
4. Convert cubic centimeters ([tex]\(\text{cm}^3\)[/tex]) to liters (L):
- Since [tex]\( 1 \text{liter} = 1000 \text{cm}^3 \)[/tex]:
[tex]\[
\text{Volume in liters} = \frac{3848.451 \text{cm}^3}{1000} \approx 3.848 \, \text{liters}
\][/tex]
Therefore, the cylindrical pipe can hold approximately [tex]\( 3.848 \)[/tex] liters of water.
