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**Topic: Calculating Surface Area/Height**

1. A cylindrical barrel (drum) contains 42 gallons of oil. The diameter of this barrel is 18 inches.

You may use the following information:
- 1 gallon = 3.78541 liters
- 1 inch = 2.54 cm
- 1 mL = 1 cm³

**Formulas:**
- Volume = π × r² × h, let π = 3.142
- Surface area of a cylinder with a closed lid and base = (2 × π × r²) + (2 × π × r × h)

1.1. Determine the radius of the barrel (drum) in centimeters.

1.2. Show, by calculations, that the height of the barrel of oil is 96.82 cm.

1.3. Calculate the surface area of this barrel in m².

Answer :

To solve the problem involving the cylindrical barrel, let's break it down into the required parts.

1.1. Determine the radius of a barrel (drum) in centimetres.

The diameter of the barrel is given as 18 inches. To find the radius:


  1. Convert the diameter from inches to centimetres:

    [tex]18 \text{ inches} \times 2.54 \text{ cm/inch} = 45.72 \text{ cm}[/tex]


  2. Calculate the radius, which is half of the diameter:

    [tex]\text{Radius} = \frac{45.72 \text{ cm}}{2} = 22.86 \text{ cm}[/tex]



1.2. Show, by calculations, that the height of the barrel is 96.82 cm.

We're provided with the volume of the barrel in gallons. First, convert this volume to cubic centimeters:


  1. Convert volume from gallons to litres:

    [tex]42 \text{ gallons} \times 3.78541 \text{ L/gallon} = 158.94922 \text{ L}[/tex]


  2. Convert litres to cubic centimeters (since 1 L = 1000 cm³):

    [tex]158.94922 \text{ L} \times 1000 \text{ cm}^3/\text{L} = 158949.22 \text{ cm}^3[/tex]


  3. Calculate the height of the cylinder using the formula for the volume of a cylinder [tex]V = \pi \times r^2 \times h[/tex]:

    [tex]158949.22 = 3.142 \times (22.86)^2 \times h[/tex]


  4. Solve for [tex]h[/tex]:

    [tex]h = \frac{158949.22}{3.142 \times 522.9796} \approx 96.82 \text{ cm}[/tex]



1.3. Calculate the surface area of this barrel in m².

Using the surface area formula for a closed cylinder, [tex]A = 2\pi r^2 + 2\pi rh[/tex]:


  1. Calculate the surface area in cm²:

    [tex]A = 2 \times 3.142 \times (22.86)^2 + 2 \times 3.142 \times 22.86 \times 96.82[/tex]


  2. Evaluate the components:


    • [tex]2 \times 3.142 \times (22.86)^2 = 3281.476[/tex]

    • [tex]2 \times 3.142 \times 22.86 \times 96.82 = 13919.0848[/tex]



  3. Total surface area in cm²:

    [tex]3281.476 + 13919.0848 = 17200.5608 \text{ cm}^2[/tex]


  4. Convert cm² to m² (since 1 m² = 10,000 cm²):

    [tex]\frac{17200.5608}{10000} = 1.72005608 \text{ m}^2[/tex]



Thus, the surface area of the barrel is approximately [tex]1.72 \text{ m}^2[/tex].

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