College

We appreciate your visit to 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. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

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 $= 3.78541$ liters
- 1 inch $= 2.54$ cm
- $1 \text{ ml} = 1 \text{ cm}^3$
- Volume $= \pi \times r^2 \times h$, let $\pi = 3.142$
- Surface area of a cylinder with a closed lid and base $= \left(2 \times \pi \times r^2\right) + (2 \times \pi \times r \times h)$

Use the information above to answer the questions that follow:

3.2.1 Determine the radius of a barrel (drum) in centimeters. (3 points)

3.2.2 Show, using calculations, that the height of the barrel of oil is $96.82$ cm. (5 points)

3.2.3 Calculate the surface area of this barrel in $m^2$.

3.3 The moisturizing gel that the hairdresser uses when relaxing hair is sold in cylindrical containers with a volume of $500 \text{ ml}$ and a radius of $4.5 \text{ cm}$.

Use the information above to answer the questions that follow:

3.3.1 The hairdresser needs to calculate the height of each container to determine how many containers she can stack on a shelf. Calculate the height using the following formula:
\[ \text{Height of a container} = \frac{\text{Volume}}{\pi \times r^2}, \text{ using } \pi = 3.14 \text{ and } 1 \text{ ml} = 1 \text{ cm}^3 \]

3.3.2 The wholesalers have a promotion on the moisturizing gel. They are now selling 600 ml of the moisturizing gel for the same price as 500 ml of the same gel. Calculate the percentage increase in the volume of the moisturizing gel using the following formula:
\[ \text{Percentage increase} = \frac{\text{new volume} - \text{original volume}}{\text{original volume}} \times 100\% \]

[Total: 33 points]

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.

Thanks for taking the time to read 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. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada