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Answer :
Let's tackle each part of the question step-by-step.
3.2.1 Determine the radius of a barrel (drum) in centimeters.
The diameter of the barrel is given as 18 inches. The radius is half of the diameter. So, the radius in inches is:
[tex]\text{Radius in inches} = \frac{18}{2} = 9 \text{ inches}[/tex]
To convert this radius to centimeters, we use the conversion factor [tex]1 \text{ inch} = 2.54 \text{ cm}[/tex]:
[tex]\text{Radius in cm} = 9 \times 2.54 = 22.86 \text{ cm}[/tex]
3.2.2 Show, using calculations, that the height of the barrel of oil is 96.82 cm.
First, find the volume in liters using the conversion [tex]1 \text{ gallon} = 3.78541 \text{ liters}[/tex]:
[tex]42 \text{ gallons} \times 3.78541 = 158.50322 \text{ liters}[/tex]
Since [tex]1 \text{ ml} = 1 \text{ cm}^3[/tex], the volume in [tex]\text{cm}^3[/tex] is:
[tex]158503.22 \text{ cm}^3[/tex]
The volume of a cylinder is calculated as:
[tex]\text{Volume} = \pi \times r^2 \times h[/tex]
Rearranging for height [tex]h[/tex]:
[tex]h = \frac{\text{Volume}}{\pi \times r^2}[/tex]
Substitute the known values:
[tex]h = \frac{158503.22}{3.142 \times (22.86)^2} \approx 96.82 \text{ cm}[/tex]
3.2.3 Calculate the surface area of this barrel in [tex]m^2[/tex].
The surface area of a cylinder is given by:
[tex]\text{Surface area} = 2 \times \pi \times r^2 + 2 \times \pi \times r \times h[/tex]
Substitute the known values:
[tex]\begin{align*}
\text{Surface area} & = 2 \times 3.142 \times (22.86)^2 + 2 \times 3.142 \times 22.86 \times 96.82 \\
& \approx 3283.87 + 13902.49 \\
& = 17186.36 \text{ cm}^2
\end{align*}[/tex]
Convert [tex]\text{cm}^2[/tex] to [tex]\text{m}^2[/tex] by dividing by 10,000:
[tex]\frac{17186.36}{10000} = 1.7186 \text{ m}^2[/tex]
3.3.1 Calculate the height of each container.
Using the formula for the height of a cylinder:
[tex]\text{Height} = \frac{\text{Volume}}{\pi \times r^2}[/tex]
Substitute the given values:
[tex]\text{Height} = \frac{500}{3.14 \times (4.5)^2} \approx 7.85 \text{ cm}[/tex]
3.3.2 Calculate the percentage increase in the volume of the moisturizing gel.
Use the formula for percentage increase:
[tex]\text{Percentage increase} = \frac{600 - 500}{500} \times 100\%[/tex]
Calculate:
[tex]\frac{100}{500} \times 100\% = 20\%[/tex]
So, the percentage increase in volume is 20%.
Each part of the problem uses basic geometry and unit conversion techniques to arrive at the solutions. Make sure to understand how unit conversions play a crucial role in accurately solving real-world problems like this one.
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