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Answer :
Below is a step-by-step solution to the problem.
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Step 1. Determine the Radius in Centimetres
The barrel has a diameter of 18 inches. First, convert the diameter to centimetres using the conversion
$$1\text{ inch} = 2.54\text{ cm}.$$
Thus, the diameter in centimetres is:
$$
\text{Diameter}_{\text{cm}} = 18 \times 2.54 = 45.72\,\text{cm}.
$$
The radius is half the diameter:
$$
r = \frac{45.72}{2} = 22.86\,\text{cm}.
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Step 2. Determine the Height of the Barrel
The barrel contains 42 gallons of oil. First, convert the volume from gallons to litres using
$$1\,\text{gallon} = 3.78541\,\text{litres},$$
and then convert the litres to cubic centimetres using
$$1\,\text{litre} = 1000\,\text{cm}^3.$$
Volume in litres:
$$
V_{\text{litres}} = 42 \times 3.78541.
$$
Volume in cubic centimetres:
$$
V_{\text{cm}^3} = V_{\text{litres}} \times 1000.
$$
The volume formula for a cylinder is given by:
$$
V = \pi r^2 h,
$$
with $\pi = 3.142$. Solving for the height $h$, we have:
$$
h = \frac{V}{\pi r^2}.
$$
Substitute the known values:
$$
h = \frac{42 \times 3.78541 \times 1000}{3.142 \times (22.86)^2}.
$$
When computed, this gives:
$$
h \approx 96.82\,\text{cm}.
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Step 3. Calculate the Surface Area of the Barrel
The surface area of a closed cylinder (which includes the top and base) is:
$$
\text{SA} = 2\pi r^2 + 2\pi r h.
$$
Substitute $r = 22.86\,\text{cm}$ and $h = 96.82\,\text{cm}$:
$$
\text{SA}_{\text{cm}^2} = 2 \times 3.142 \times (22.86)^2 + 2 \times 3.142 \times 22.86 \times 96.82.
$$
After calculating, the surface area in square centimetres is found. To convert it to square metres note that:
$$
1\,\text{m}^2 = 10\,000\,\text{cm}^2.
$$
Thus, the surface area in square metres is:
$$
\text{SA}_{\text{m}^2} \approx 1.719\,\text{m}^2.
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Summary of Answers
1. The radius of the barrel is
$$
22.86\,\text{cm}.
$$
2. The height of the barrel is
$$
96.82\,\text{cm}.
$$
3. The surface area of the barrel is approximately
$$
1.719\,\text{m}^2.
$$
This completes the solution to the problem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Step 1. Determine the Radius in Centimetres
The barrel has a diameter of 18 inches. First, convert the diameter to centimetres using the conversion
$$1\text{ inch} = 2.54\text{ cm}.$$
Thus, the diameter in centimetres is:
$$
\text{Diameter}_{\text{cm}} = 18 \times 2.54 = 45.72\,\text{cm}.
$$
The radius is half the diameter:
$$
r = \frac{45.72}{2} = 22.86\,\text{cm}.
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Step 2. Determine the Height of the Barrel
The barrel contains 42 gallons of oil. First, convert the volume from gallons to litres using
$$1\,\text{gallon} = 3.78541\,\text{litres},$$
and then convert the litres to cubic centimetres using
$$1\,\text{litre} = 1000\,\text{cm}^3.$$
Volume in litres:
$$
V_{\text{litres}} = 42 \times 3.78541.
$$
Volume in cubic centimetres:
$$
V_{\text{cm}^3} = V_{\text{litres}} \times 1000.
$$
The volume formula for a cylinder is given by:
$$
V = \pi r^2 h,
$$
with $\pi = 3.142$. Solving for the height $h$, we have:
$$
h = \frac{V}{\pi r^2}.
$$
Substitute the known values:
$$
h = \frac{42 \times 3.78541 \times 1000}{3.142 \times (22.86)^2}.
$$
When computed, this gives:
$$
h \approx 96.82\,\text{cm}.
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Step 3. Calculate the Surface Area of the Barrel
The surface area of a closed cylinder (which includes the top and base) is:
$$
\text{SA} = 2\pi r^2 + 2\pi r h.
$$
Substitute $r = 22.86\,\text{cm}$ and $h = 96.82\,\text{cm}$:
$$
\text{SA}_{\text{cm}^2} = 2 \times 3.142 \times (22.86)^2 + 2 \times 3.142 \times 22.86 \times 96.82.
$$
After calculating, the surface area in square centimetres is found. To convert it to square metres note that:
$$
1\,\text{m}^2 = 10\,000\,\text{cm}^2.
$$
Thus, the surface area in square metres is:
$$
\text{SA}_{\text{m}^2} \approx 1.719\,\text{m}^2.
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Summary of Answers
1. The radius of the barrel is
$$
22.86\,\text{cm}.
$$
2. The height of the barrel is
$$
96.82\,\text{cm}.
$$
3. The surface area of the barrel is approximately
$$
1.719\,\text{m}^2.
$$
This completes the solution to the problem.
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