We appreciate your visit to 3 2 Crude oil is sold in barrels A cylindrical barrel drum contains 42 gallons of oil The diameter of this barrel is 18 inches. 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!
Answer :
Below is a step‐by‐step solution with all the calculations clearly shown.
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Step 3.2.1 – Finding the Radius of the Barrel in Centimetres
1. The diameter is given as 18 inches, so the radius in inches is
$$
\text{radius (in inches)} = \frac{18}{2} = 9 \text{ inches}.
$$
2. Convert the radius to centimetres using
$$
1 \text{ inch} = 2.54 \text{ cm}.
$$
Thus,
$$
\text{radius (in cm)} = 9 \times 2.54 = 22.86 \text{ cm}.
$$
────────────────────────────
Step 3.2.2 – Determining the Height of the Barrier
1. The barrel holds 42 gallons. Converting gallons to cubic centimetres:
- First, convert gallons to litres using
$$
1 \text{ gallon} = 3.78541 \text{ litres}.
$$
- Since $1 \text{ litre} = 1000 \text{ mL}$ and $1 \text{ mL} = 1 \text{ cm}^3$, then
$$
1 \text{ gallon} = 3785.41 \text{ cm}^3.
$$
- Thus, the volume of the barrel is
$$
V = 42 \times 3785.41 = 158987.22 \text{ cm}^3.
$$
2. The volume of a cylinder is given by
$$
V = \pi r^2 h.
$$
3. Solving for the height ($h$),
$$
h = \frac{V}{\pi r^2}.
$$
4. Using the values:
- $V = 158987.22 \text{ cm}^3$,
- $r = 22.86 \text{ cm}$, and
- $\pi = 3.142$,
substitute into the formula:
$$
h = \frac{158987.22}{3.142 \times (22.86)^2}.
$$
5. After computing the denominator,
$$
(22.86)^2 \approx 521.98,
$$
and then
$$
3.142 \times 521.98 \approx 1639,
$$
so
$$
h \approx \frac{158987.22}{1639} \approx 96.83 \text{ cm}.
$$
Thus, the height is approximately $96.82 \text{ cm}$.
────────────────────────────
Step 3.2.3 – Calculating the Surface Area of the Barrel in $m^2$
1. For a cylinder with a closed top and bottom (a lid and base), the surface area is given by:
$$
\text{Surface Area} = 2 \pi r^2 + 2 \pi r h.
$$
2. Using:
- $r = 22.86 \text{ cm}$,
- $h \approx 96.83 \text{ cm}$, and
- $\pi = 3.142$,
calculate each term:
- Area of both the top and the base:
$$
2 \pi r^2 = 2 \times 3.142 \times (22.86)^2.
$$
- Lateral surface area:
$$
2 \pi r h = 2 \times 3.142 \times 22.86 \times 96.83.
$$
3. The total surface area calculated comes out to approximately
$$
17193.53 \text{ cm}^2.
$$
4. Converting into square metres, recall that
$$
1\,m^2 = 10000\,cm^2,
$$
so
$$
\text{Surface Area in } m^2 = \frac{17193.53}{10000} \approx 1.71935 \, m^2.
$$
────────────────────────────
Step 3.3.1 – Calculating the Height of the Moisturising Gel Container
1. The volume of the container is $500\,m\ell$, which is equivalent to $500\,cm^3$, and the radius is given as $4.5\,cm$.
2. The volume of a cylinder is given by:
$$
V = \pi r^2 h.
$$
3. Rearranging to find the height ($h$):
$$
h = \frac{V}{\pi r^2}.
$$
4. Substitute the values:
- $V = 500\;cm^3$,
- $r = 4.5\;cm$, and
- $\pi = 3.14$:
$$
h = \frac{500}{3.14 \times (4.5)^2}.
$$
5. Notice that $(4.5)^2 = 20.25$, thus:
$$
\pi r^2 = 3.14 \times 20.25 \approx 63.585.
$$
6. Then,
$$
h \approx \frac{500}{63.585} \approx 7.86349 \text{ cm}.
$$
────────────────────────────
Step 3.3.2 – Calculating the Percentage Increase in Volume of the Gel
1. The promotion states that $600\,m\ell$ of the gel is sold for the same price as $500\,m\ell$.
2. The percentage increase is calculated using:
$$
\text{Percentage Increase} = \frac{\text{New volume} - \text{Original volume}}{\text{Original volume}} \times 100\%.
$$
3. Substitute the given volumes:
$$
\text{Percentage Increase} = \frac{600 - 500}{500} \times 100\% = \frac{100}{500} \times 100\% = 20\%.
$$
────────────────────────────
Final Answers:
- 3.2.1 The radius of the barrel is approximately $22.86\text{ cm}$.
- 3.2.2 The height of the barrel is approximately $96.82\text{ cm}$.
- 3.2.3 The surface area of the barrel is approximately $1.71935\text{ m}^2$.
- 3.3.1 The height of the moisturising gel container is approximately $7.86349\text{ cm}$.
- 3.3.2 The percentage increase in the volume of the moisturising gel is $20\%$.
────────────────────────────
Step 3.2.1 – Finding the Radius of the Barrel in Centimetres
1. The diameter is given as 18 inches, so the radius in inches is
$$
\text{radius (in inches)} = \frac{18}{2} = 9 \text{ inches}.
$$
2. Convert the radius to centimetres using
$$
1 \text{ inch} = 2.54 \text{ cm}.
$$
Thus,
$$
\text{radius (in cm)} = 9 \times 2.54 = 22.86 \text{ cm}.
$$
────────────────────────────
Step 3.2.2 – Determining the Height of the Barrier
1. The barrel holds 42 gallons. Converting gallons to cubic centimetres:
- First, convert gallons to litres using
$$
1 \text{ gallon} = 3.78541 \text{ litres}.
$$
- Since $1 \text{ litre} = 1000 \text{ mL}$ and $1 \text{ mL} = 1 \text{ cm}^3$, then
$$
1 \text{ gallon} = 3785.41 \text{ cm}^3.
$$
- Thus, the volume of the barrel is
$$
V = 42 \times 3785.41 = 158987.22 \text{ cm}^3.
$$
2. The volume of a cylinder is given by
$$
V = \pi r^2 h.
$$
3. Solving for the height ($h$),
$$
h = \frac{V}{\pi r^2}.
$$
4. Using the values:
- $V = 158987.22 \text{ cm}^3$,
- $r = 22.86 \text{ cm}$, and
- $\pi = 3.142$,
substitute into the formula:
$$
h = \frac{158987.22}{3.142 \times (22.86)^2}.
$$
5. After computing the denominator,
$$
(22.86)^2 \approx 521.98,
$$
and then
$$
3.142 \times 521.98 \approx 1639,
$$
so
$$
h \approx \frac{158987.22}{1639} \approx 96.83 \text{ cm}.
$$
Thus, the height is approximately $96.82 \text{ cm}$.
────────────────────────────
Step 3.2.3 – Calculating the Surface Area of the Barrel in $m^2$
1. For a cylinder with a closed top and bottom (a lid and base), the surface area is given by:
$$
\text{Surface Area} = 2 \pi r^2 + 2 \pi r h.
$$
2. Using:
- $r = 22.86 \text{ cm}$,
- $h \approx 96.83 \text{ cm}$, and
- $\pi = 3.142$,
calculate each term:
- Area of both the top and the base:
$$
2 \pi r^2 = 2 \times 3.142 \times (22.86)^2.
$$
- Lateral surface area:
$$
2 \pi r h = 2 \times 3.142 \times 22.86 \times 96.83.
$$
3. The total surface area calculated comes out to approximately
$$
17193.53 \text{ cm}^2.
$$
4. Converting into square metres, recall that
$$
1\,m^2 = 10000\,cm^2,
$$
so
$$
\text{Surface Area in } m^2 = \frac{17193.53}{10000} \approx 1.71935 \, m^2.
$$
────────────────────────────
Step 3.3.1 – Calculating the Height of the Moisturising Gel Container
1. The volume of the container is $500\,m\ell$, which is equivalent to $500\,cm^3$, and the radius is given as $4.5\,cm$.
2. The volume of a cylinder is given by:
$$
V = \pi r^2 h.
$$
3. Rearranging to find the height ($h$):
$$
h = \frac{V}{\pi r^2}.
$$
4. Substitute the values:
- $V = 500\;cm^3$,
- $r = 4.5\;cm$, and
- $\pi = 3.14$:
$$
h = \frac{500}{3.14 \times (4.5)^2}.
$$
5. Notice that $(4.5)^2 = 20.25$, thus:
$$
\pi r^2 = 3.14 \times 20.25 \approx 63.585.
$$
6. Then,
$$
h \approx \frac{500}{63.585} \approx 7.86349 \text{ cm}.
$$
────────────────────────────
Step 3.3.2 – Calculating the Percentage Increase in Volume of the Gel
1. The promotion states that $600\,m\ell$ of the gel is sold for the same price as $500\,m\ell$.
2. The percentage increase is calculated using:
$$
\text{Percentage Increase} = \frac{\text{New volume} - \text{Original volume}}{\text{Original volume}} \times 100\%.
$$
3. Substitute the given volumes:
$$
\text{Percentage Increase} = \frac{600 - 500}{500} \times 100\% = \frac{100}{500} \times 100\% = 20\%.
$$
────────────────────────────
Final Answers:
- 3.2.1 The radius of the barrel is approximately $22.86\text{ cm}$.
- 3.2.2 The height of the barrel is approximately $96.82\text{ cm}$.
- 3.2.3 The surface area of the barrel is approximately $1.71935\text{ m}^2$.
- 3.3.1 The height of the moisturising gel container is approximately $7.86349\text{ cm}$.
- 3.3.2 The percentage increase in the volume of the moisturising gel is $20\%$.
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