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Answer :
We wish to determine several characteristics of a cylindrical barrel that contains 42 gallons of oil and has a diameter of 18 inches. We use the following information:
• 1 gallon = 3.78541 litres
• 1 inch = 2.54 cm
• 1 ml = 1 cm³ (and 1 litre = 1000 cm³)
• Volume of a cylinder: $$V = \pi r^2 h$$ (with $$\pi = 3.142$$)
• Surface area of a closed cylinder (both top and bottom):
$$A = 2\pi r^2 + 2\pi r h$$
Below is the step‐by-step calculation.
────────────────────────────
Step 1: Determine the Radius in Centimetres
The barrel’s diameter is 18 inches. Therefore, the radius in inches is
$$
r_{\text{inch}} = \frac{18}{2} = 9 \text{ inches}.
$$
Converting to centimetres using $$1 \text{ inch} = 2.54 \text{ cm}$$:
$$
r = 9 \times 2.54 = 22.86 \text{ cm}.
$$
────────────────────────────
Step 2: Determine the Height of the Barrel
First, we convert the volume of oil from gallons to cubic centimetres.
1. Convert gallons to litres:
$$
\text{Volume in litres} = 42 \times 3.78541.
$$
This gives approximately
$$
158.98722 \text{ litres}.
$$
2. Convert litres to cubic centimetres:
$$
\text{Volume in cm}^3 = 158.98722 \times 1000 \approx 158987.22 \text{ cm}^3.
$$
Now, using the formula for the volume of a cylinder,
$$
V = \pi r^2 h,
$$
we can solve for the height $$h$$ by rearranging the formula:
$$
h = \frac{V}{\pi r^2}.
$$
Substitute the known values ($$V \approx 158987.22 \text{ cm}^3$$, $$r = 22.86 \text{ cm}$$, and $$\pi = 3.142$$):
$$
h = \frac{158987.22}{3.142 \times (22.86)^2}.
$$
Calculating the denominator:
$$
r^2 = (22.86)^2 \approx 522.99,
$$
so
$$
\pi r^2 \approx 3.142 \times 522.99 \approx 1643.94.
$$
Thus, the height is approximately
$$
h \approx \frac{158987.22}{1643.94} \approx 96.83 \text{ cm}.
$$
Rounded to two decimal places, the height is $$96.82 \text{ cm}$$.
────────────────────────────
Step 3: Calculate the Surface Area of the Barrel in Square Metres
The surface area of a closed cylinder (including the top and bottom) is given by
$$
A = 2\pi r^2 + 2\pi r h.
$$
Substitute the known values: $$r = 22.86 \text{ cm}$$, $$h \approx 96.82 \text{ cm}$$, and $$\pi = 3.142$$.
1. Calculate the area of the top and bottom:
$$
2\pi r^2 = 2 \times 3.142 \times (22.86)^2.
$$
We have
$$
(22.86)^2 \approx 522.99,
$$
so
$$
2\pi r^2 \approx 2 \times 3.142 \times 522.99 \approx 3287.88 \text{ cm}^2.
$$
2. Calculate the lateral surface area:
$$
2\pi r h = 2 \times 3.142 \times 22.86 \times 96.82.
$$
This gives approximately
$$
2\pi r h \approx 13905.65 \text{ cm}^2.
$$
Now, the total surface area in square centimetres is
$$
A \approx 3287.88 + 13905.65 \approx 17193.53 \text{ cm}^2.
$$
Finally, convert the area to square metres knowing that $$1 \text{ m}^2 = 10000 \text{ cm}^2$$:
$$
A \approx \frac{17193.53}{10000} \approx 1.71935 \text{ m}^2.
$$
────────────────────────────
Summary of Answers
1. The radius of the barrel in centimetres is
$$22.86 \text{ cm}.$$
2. The height of the barrel is approximately
$$96.82 \text{ cm}.$$
3. The surface area of the barrel is approximately
$$1.719 \text{ m}^2.$$
These calculations provide the required dimensions and surface area of the barrel.
• 1 gallon = 3.78541 litres
• 1 inch = 2.54 cm
• 1 ml = 1 cm³ (and 1 litre = 1000 cm³)
• Volume of a cylinder: $$V = \pi r^2 h$$ (with $$\pi = 3.142$$)
• Surface area of a closed cylinder (both top and bottom):
$$A = 2\pi r^2 + 2\pi r h$$
Below is the step‐by-step calculation.
────────────────────────────
Step 1: Determine the Radius in Centimetres
The barrel’s diameter is 18 inches. Therefore, the radius in inches is
$$
r_{\text{inch}} = \frac{18}{2} = 9 \text{ inches}.
$$
Converting to centimetres using $$1 \text{ inch} = 2.54 \text{ cm}$$:
$$
r = 9 \times 2.54 = 22.86 \text{ cm}.
$$
────────────────────────────
Step 2: Determine the Height of the Barrel
First, we convert the volume of oil from gallons to cubic centimetres.
1. Convert gallons to litres:
$$
\text{Volume in litres} = 42 \times 3.78541.
$$
This gives approximately
$$
158.98722 \text{ litres}.
$$
2. Convert litres to cubic centimetres:
$$
\text{Volume in cm}^3 = 158.98722 \times 1000 \approx 158987.22 \text{ cm}^3.
$$
Now, using the formula for the volume of a cylinder,
$$
V = \pi r^2 h,
$$
we can solve for the height $$h$$ by rearranging the formula:
$$
h = \frac{V}{\pi r^2}.
$$
Substitute the known values ($$V \approx 158987.22 \text{ cm}^3$$, $$r = 22.86 \text{ cm}$$, and $$\pi = 3.142$$):
$$
h = \frac{158987.22}{3.142 \times (22.86)^2}.
$$
Calculating the denominator:
$$
r^2 = (22.86)^2 \approx 522.99,
$$
so
$$
\pi r^2 \approx 3.142 \times 522.99 \approx 1643.94.
$$
Thus, the height is approximately
$$
h \approx \frac{158987.22}{1643.94} \approx 96.83 \text{ cm}.
$$
Rounded to two decimal places, the height is $$96.82 \text{ cm}$$.
────────────────────────────
Step 3: Calculate the Surface Area of the Barrel in Square Metres
The surface area of a closed cylinder (including the top and bottom) is given by
$$
A = 2\pi r^2 + 2\pi r h.
$$
Substitute the known values: $$r = 22.86 \text{ cm}$$, $$h \approx 96.82 \text{ cm}$$, and $$\pi = 3.142$$.
1. Calculate the area of the top and bottom:
$$
2\pi r^2 = 2 \times 3.142 \times (22.86)^2.
$$
We have
$$
(22.86)^2 \approx 522.99,
$$
so
$$
2\pi r^2 \approx 2 \times 3.142 \times 522.99 \approx 3287.88 \text{ cm}^2.
$$
2. Calculate the lateral surface area:
$$
2\pi r h = 2 \times 3.142 \times 22.86 \times 96.82.
$$
This gives approximately
$$
2\pi r h \approx 13905.65 \text{ cm}^2.
$$
Now, the total surface area in square centimetres is
$$
A \approx 3287.88 + 13905.65 \approx 17193.53 \text{ cm}^2.
$$
Finally, convert the area to square metres knowing that $$1 \text{ m}^2 = 10000 \text{ cm}^2$$:
$$
A \approx \frac{17193.53}{10000} \approx 1.71935 \text{ m}^2.
$$
────────────────────────────
Summary of Answers
1. The radius of the barrel in centimetres is
$$22.86 \text{ cm}.$$
2. The height of the barrel is approximately
$$96.82 \text{ cm}.$$
3. The surface area of the barrel is approximately
$$1.719 \text{ m}^2.$$
These calculations provide the required dimensions and surface area of the barrel.
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