High School

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Crude oil is sold in barrels. 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 litres
- 1 inch = 2.54 cm
- 1 ml = 1 cm³

Formulas:
- Volume = [tex]\pi \times r^2 \times h[/tex], let [tex]\pi = 3.142[/tex]
- Surface area of a cylinder with a closed lid and base = [tex](2 \times \pi \times r^2) + (2 \times \pi \times r \times h)[/tex]

Use the information above to answer the questions that follow:

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

3.2.2 Show, using calculations, that the height of the barrel of oil is 96.82 cm.

3.2.3 Calculate the surface area of this barrel in [tex]m^2[/tex].

Answer :

To solve this problem, let's break it down step-by-step.

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

The diameter of the barrel is given as 18 inches. First, we'll convert this to centimetres using the conversion factor given (1 inch = 2.54 cm).

[tex]\text{Diameter in cm} = 18 \times 2.54 = 45.72 \text{ cm}[/tex]

The radius is half of the diameter. Therefore:

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

3.2.2 Show using calculations, that the height of the barrel of oil is 96.82 cm.

We want to find the height of the cylinder when it holds 42 gallons of oil. Let's convert the volume from gallons to litres and then to cubic centimeters (as 1 ml = 1 cm³):

  1. Convert 42 gallons to litres:

[tex]42 \times 3.78541 = 158.98722 \text{ litres}[/tex]

  1. Convert litres to cubic centimetres (since 1 litre = 1000 cm³):

[tex]158.98722 \times 1000 = 158987.22 \text{ cm}^3[/tex]

Now, use the formula for the volume of a cylinder:

[tex]V = \pi \times r^2 \times h[/tex]

Substitute the known values to find the height [tex]h[/tex]:

[tex]158987.22 = 3.142 \times (22.86)^2 \times h[/tex]

Solving for [tex]h[/tex]:

[tex]158987.22 = 3.142 \times 522.0996 \times h[/tex]

[tex]158987.22 = 1640.9709352 \times h[/tex]

[tex]h = \frac{158987.22}{1640.9709352} \approx 96.82 \text{ cm}[/tex]

Thus, the height of the barrel is confirmed to be 96.82 cm.

3.2.3 Calculate the surface area of this barrel in m².

The formula for the surface area of a cylinder with closed lid and base is:

[tex]\text{Surface Area} = 2 \times \pi \times r^2 + 2 \times \pi \times r \times h[/tex]

Substitute the known values:

[tex]\text{Surface Area} = 2 \times 3.142 \times (22.86)^2 + 2 \times 3.142 \times 22.86 \times 96.82[/tex]

Calculate each component:

  1. Top and bottom area:

[tex]2 \times 3.142 \times 522.0996 = 3278.6225[/tex]

  1. Lateral area:

[tex]2 \times 3.142 \times 22.86 \times 96.82 = 13921.0273[/tex]

Add them together:

[tex]3278.6225 + 13921.0273 = 17199.6498 \text{ cm}^2[/tex]

Convert [tex]\text{cm}^2[/tex] to [tex]\text{m}^2[/tex] by dividing by 10000:

[tex]\text{Surface Area in } \text{m}^2 = \frac{17199.6498}{10000} \approx 1.72 \text{ m}^2[/tex]

Therefore, the surface area of the barrel is approximately 1.72 m².

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Rewritten by : Barada