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Answer :
- The radius of the barrel is calculated by dividing the diameter by 2 and converting inches to centimeters: $r = 22.86 \text{ cm}$.
- The height is found using the volume formula for a cylinder, converting gallons to cubic centimeters: $h \approx 96.83 \text{ cm}$.
- The surface area of the barrel is calculated using the formula $2 \pi r^2$ and converted to square meters: $SA \approx 0.328 \text{ m}^2$.
- The final answers are: radius = $\boxed{22.86 \text{ cm}}$, height $\approx$ $\boxed{96.83 \text{ cm}}$, surface area $\approx$ $\boxed{0.328 \text{ m}^2}$.
### Explanation
1. Calculate the radius in cm
First, let's calculate the radius of the barrel in centimeters. We know the diameter is 18 inches, so the radius is half of that. Then, we convert inches to centimeters using the given conversion factor.
2. Radius calculation
The radius in inches is:
$$r_{inches} = \frac{18}{2} = 9 \text{ inches}$$
Now, convert inches to centimeters:
$$r_{cm} = 9 \text{ inches} \times 2.54 \frac{\text{cm}}{\text{inch}} = 22.86 \text{ cm}$$
So, the radius of the barrel is 22.86 cm.
3. Calculate the height in cm
Next, we need to show that the height of the barrel is approximately 96 cm. We'll use the volume formula for a cylinder and the given conversions to find the height.
4. Height calculation
First, convert the volume from gallons to cubic centimeters:
$$V_{gallons} = 42 \text{ gallons}$$
$$V_{litres} = 42 \text{ gallons} \times 3.78541 \frac{\text{litres}}{\text{gallon}} = 158.98722 \text{ litres}$$
$$V_{cm^3} = 158.98722 \text{ litres} \times 1000 \frac{\text{cm}^3}{\text{litre}} = 158987.22 \text{ cm}^3$$
Now, use the volume formula for a cylinder to solve for the height:
$$V = \pi r^2 h$$
$$h = \frac{V}{\pi r^2} = \frac{158987.22}{3.142 \times (22.86)^2} = \frac{158987.22}{3.142 \times 522.5796} = \frac{158987.22}{1642.02} \approx 96.83 \text{ cm}$$
Therefore, the height of the barrel is approximately 96.83 cm, which is close to 96 cm.
5. Calculate the surface area in $m^2$
Finally, we need to calculate the surface area of the barrel in $m^2$. We'll use the formula for the surface area of a closed cylinder and the calculated radius.
6. Surface area calculation
The surface area of the barrel in $cm^2$ is:
$$SA = 2 \pi r^2 = 2 \times 3.142 \times (22.86)^2 = 2 \times 3.142 \times 522.5796 \approx 3283.89 \text{ cm}^2$$
Now, convert the surface area from $cm^2$ to $m^2$:
$$SA_{m^2} = \frac{3283.89 \text{ cm}^2}{10000 \frac{\text{cm}^2}{\text{m}^2}} = 0.328389 \text{ m}^2$$
So, the surface area of the barrel is approximately 0.328 $m^2$.
7. Final Answer
In summary:
- The radius of the barrel is 22.86 cm.
- The height of the barrel is approximately 96.83 cm, which is close to 96 cm.
- The surface area of the barrel is approximately 0.328 $m^2$.
### Examples
Understanding the dimensions and surface area of barrels is crucial in industries like oil and gas for storage and transportation. Knowing the surface area helps in calculating the amount of coating needed to protect the barrel from corrosion, while accurate volume and height measurements ensure efficient storage and transportation logistics. This also applies to the food and beverage industry, where similar cylindrical containers are used.
- The height is found using the volume formula for a cylinder, converting gallons to cubic centimeters: $h \approx 96.83 \text{ cm}$.
- The surface area of the barrel is calculated using the formula $2 \pi r^2$ and converted to square meters: $SA \approx 0.328 \text{ m}^2$.
- The final answers are: radius = $\boxed{22.86 \text{ cm}}$, height $\approx$ $\boxed{96.83 \text{ cm}}$, surface area $\approx$ $\boxed{0.328 \text{ m}^2}$.
### Explanation
1. Calculate the radius in cm
First, let's calculate the radius of the barrel in centimeters. We know the diameter is 18 inches, so the radius is half of that. Then, we convert inches to centimeters using the given conversion factor.
2. Radius calculation
The radius in inches is:
$$r_{inches} = \frac{18}{2} = 9 \text{ inches}$$
Now, convert inches to centimeters:
$$r_{cm} = 9 \text{ inches} \times 2.54 \frac{\text{cm}}{\text{inch}} = 22.86 \text{ cm}$$
So, the radius of the barrel is 22.86 cm.
3. Calculate the height in cm
Next, we need to show that the height of the barrel is approximately 96 cm. We'll use the volume formula for a cylinder and the given conversions to find the height.
4. Height calculation
First, convert the volume from gallons to cubic centimeters:
$$V_{gallons} = 42 \text{ gallons}$$
$$V_{litres} = 42 \text{ gallons} \times 3.78541 \frac{\text{litres}}{\text{gallon}} = 158.98722 \text{ litres}$$
$$V_{cm^3} = 158.98722 \text{ litres} \times 1000 \frac{\text{cm}^3}{\text{litre}} = 158987.22 \text{ cm}^3$$
Now, use the volume formula for a cylinder to solve for the height:
$$V = \pi r^2 h$$
$$h = \frac{V}{\pi r^2} = \frac{158987.22}{3.142 \times (22.86)^2} = \frac{158987.22}{3.142 \times 522.5796} = \frac{158987.22}{1642.02} \approx 96.83 \text{ cm}$$
Therefore, the height of the barrel is approximately 96.83 cm, which is close to 96 cm.
5. Calculate the surface area in $m^2$
Finally, we need to calculate the surface area of the barrel in $m^2$. We'll use the formula for the surface area of a closed cylinder and the calculated radius.
6. Surface area calculation
The surface area of the barrel in $cm^2$ is:
$$SA = 2 \pi r^2 = 2 \times 3.142 \times (22.86)^2 = 2 \times 3.142 \times 522.5796 \approx 3283.89 \text{ cm}^2$$
Now, convert the surface area from $cm^2$ to $m^2$:
$$SA_{m^2} = \frac{3283.89 \text{ cm}^2}{10000 \frac{\text{cm}^2}{\text{m}^2}} = 0.328389 \text{ m}^2$$
So, the surface area of the barrel is approximately 0.328 $m^2$.
7. Final Answer
In summary:
- The radius of the barrel is 22.86 cm.
- The height of the barrel is approximately 96.83 cm, which is close to 96 cm.
- The surface area of the barrel is approximately 0.328 $m^2$.
### Examples
Understanding the dimensions and surface area of barrels is crucial in industries like oil and gas for storage and transportation. Knowing the surface area helps in calculating the amount of coating needed to protect the barrel from corrosion, while accurate volume and height measurements ensure efficient storage and transportation logistics. This also applies to the food and beverage industry, where similar cylindrical containers are used.
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