We appreciate your visit to Capricorn South May 2025 Mathematical Literacy Grade 12 Problem Crude oil is sold in barrels A cylindrical barrel drum contains 42 gallons of oil The. 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 :
We start by noting that the barrel is a cylinder with a diameter of 18 inches, and it holds 42 gallons of oil.
------------------------------------------
Step 1. Determine the radius of the barrel in centimeters.
The diameter of the barrel is given as 18 inches. The radius in inches is half of the diameter:
$$
r_{\text{in}} = \frac{18}{2} = 9 \text{ inches}.
$$
We convert inches to centimeters using the conversion factor $1 \text{ inch} = 2.54 \text{ cm}$:
$$
r_{\text{cm}} = 9 \times 2.54 = 22.86 \text{ cm}.
$$
------------------------------------------
Step 2. Calculate the height of the barrel given that it holds 42 gallons of oil.
First, convert the volume from gallons to cubic centimeters. We are given:
- $1 \text{ gallon} = 3.78541 \text{ litres}$,
- $1 \text{ litre} = 1000 \text{ cm}^3$.
Thus, the volume in cubic centimeters is:
$$
V = 42 \times 3.78541 \times 1000 = 158987.22 \text{ cm}^3.
$$
The volume of a cylinder is given by:
$$
V = \pi r^2 h,
$$
where we use $\pi = 3.142$. Solving for the height $h$, we have:
$$
h = \frac{V}{\pi r^2}.
$$
Substitute the known values:
$$
h = \frac{158987.22}{3.142 \times (22.86)^2}.
$$
Calculating the denominator:
$$
r^2 = (22.86)^2 \approx 522.59,
$$
$$
\pi r^2 \approx 3.142 \times 522.59 \approx 1642.08.
$$
Thus, the height is:
$$
h \approx \frac{158987.22}{1642.08} \approx 96.82 \text{ cm}.
$$
------------------------------------------
Step 3. Calculate the surface area of the barrel in square meters.
The surface area $S$ of a closed cylinder (with a top and bottom) is given by:
$$
S = 2\pi r^2 + 2\pi r h.
$$
Substitute $r = 22.86 \text{ cm}$, $h = 96.82 \text{ cm}$, and $\pi = 3.142$:
1. Calculate the area of the two circular bases:
$$
2\pi r^2 = 2 \times 3.142 \times (22.86)^2.
$$
With $r^2 \approx 522.59$, we get:
$$
2\pi r^2 \approx 2 \times 3.142 \times 522.59 \approx 3284.16 \text{ cm}^2.
$$
2. Calculate the lateral (curved) surface area:
$$
2\pi r h = 2 \times 3.142 \times 22.86 \times 96.82.
$$
This gives:
$$
2\pi r h \approx 2 \times 3.142 \times 22.86 \times 96.82 \approx 13909.37 \text{ cm}^2.
$$
Adding these parts:
$$
\text{Total Surface Area} = 3284.16 + 13909.37 \approx 17193.53 \text{ cm}^2.
$$
To convert from square centimeters to square meters, note that $1 \text{ m}^2 = 10,000 \text{ cm}^2$:
$$
S_{\text{m}^2} = \frac{17193.53}{10000} \approx 1.71935 \text{ m}^2.
$$
------------------------------------------
Final Answers:
1. The radius of the barrel is
$$
22.86 \text{ cm}.
$$
2. The height of the barrel is
$$
96.82 \text{ cm}.
$$
3. The surface area of the barrel is
$$
1.71935 \text{ m}^2.
$$
------------------------------------------
Step 1. Determine the radius of the barrel in centimeters.
The diameter of the barrel is given as 18 inches. The radius in inches is half of the diameter:
$$
r_{\text{in}} = \frac{18}{2} = 9 \text{ inches}.
$$
We convert inches to centimeters using the conversion factor $1 \text{ inch} = 2.54 \text{ cm}$:
$$
r_{\text{cm}} = 9 \times 2.54 = 22.86 \text{ cm}.
$$
------------------------------------------
Step 2. Calculate the height of the barrel given that it holds 42 gallons of oil.
First, convert the volume from gallons to cubic centimeters. We are given:
- $1 \text{ gallon} = 3.78541 \text{ litres}$,
- $1 \text{ litre} = 1000 \text{ cm}^3$.
Thus, the volume in cubic centimeters is:
$$
V = 42 \times 3.78541 \times 1000 = 158987.22 \text{ cm}^3.
$$
The volume of a cylinder is given by:
$$
V = \pi r^2 h,
$$
where we use $\pi = 3.142$. Solving for the height $h$, we have:
$$
h = \frac{V}{\pi r^2}.
$$
Substitute the known values:
$$
h = \frac{158987.22}{3.142 \times (22.86)^2}.
$$
Calculating the denominator:
$$
r^2 = (22.86)^2 \approx 522.59,
$$
$$
\pi r^2 \approx 3.142 \times 522.59 \approx 1642.08.
$$
Thus, the height is:
$$
h \approx \frac{158987.22}{1642.08} \approx 96.82 \text{ cm}.
$$
------------------------------------------
Step 3. Calculate the surface area of the barrel in square meters.
The surface area $S$ of a closed cylinder (with a top and bottom) is given by:
$$
S = 2\pi r^2 + 2\pi r h.
$$
Substitute $r = 22.86 \text{ cm}$, $h = 96.82 \text{ cm}$, and $\pi = 3.142$:
1. Calculate the area of the two circular bases:
$$
2\pi r^2 = 2 \times 3.142 \times (22.86)^2.
$$
With $r^2 \approx 522.59$, we get:
$$
2\pi r^2 \approx 2 \times 3.142 \times 522.59 \approx 3284.16 \text{ cm}^2.
$$
2. Calculate the lateral (curved) surface area:
$$
2\pi r h = 2 \times 3.142 \times 22.86 \times 96.82.
$$
This gives:
$$
2\pi r h \approx 2 \times 3.142 \times 22.86 \times 96.82 \approx 13909.37 \text{ cm}^2.
$$
Adding these parts:
$$
\text{Total Surface Area} = 3284.16 + 13909.37 \approx 17193.53 \text{ cm}^2.
$$
To convert from square centimeters to square meters, note that $1 \text{ m}^2 = 10,000 \text{ cm}^2$:
$$
S_{\text{m}^2} = \frac{17193.53}{10000} \approx 1.71935 \text{ m}^2.
$$
------------------------------------------
Final Answers:
1. The radius of the barrel is
$$
22.86 \text{ cm}.
$$
2. The height of the barrel is
$$
96.82 \text{ cm}.
$$
3. The surface area of the barrel is
$$
1.71935 \text{ m}^2.
$$
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