Calculating Surface Area Increase of a Bisected Cylinder

The problem asks us to find the percentage increase in the total surface area when a solid cylinder is bisected (cut into two equal halves) along its height.

1. Understand the Original Cylinder

We are given the dimensions of the original solid cylinder:

• Radius of the base (r) = 28 cm
• Height (h) = 24 cm

The formula for the total surface area (TSA) of a solid cylinder is:

\( \text{TSA}_{\text{original}} = 2\pi r(r+h) = 2\pi r^2 + 2\pi r h \)

Let's calculate the initial total surface area using the given values:

\( \text{TSA}_{\text{original}} = 2\pi (28 \text{ cm})(28 \text{ cm} + 24 \text{ cm}) \)

\( \text{TSA}_{\text{original}} = 2\pi (28 \text{ cm})(52 \text{ cm}) \)

\( \text{TSA}_{\text{original}} = 2\pi (1456 \text{ cm}^2) \)

\( \text{TSA}_{\text{original}} = 2912\pi \text{ cm}^2 \)

2. Analyze the Bisected Cylinders

When the cylinder is bisected from its height, it is cut horizontally through the middle. This results in two identical smaller cylinders.

• Radius of each new cylinder (r') = original radius (r) = 28 cm
• Height of each new cylinder (h') = original height (h) / 2 = 24 cm / 2 = 12 cm

When the cut is made, a new circular surface is exposed on each of the two resulting pieces. The area of this new surface is \(\pi r^2\).

3. Calculate Total Surface Area After Bisection

Each of the two new cylinders has its own total surface area. The formula for the total surface area of one of these smaller cylinders (with height h' and radius r') is the standard formula, but we must account for the surfaces present:

Each new cylinder has:

• One original base (area = \(\pi r^2\))
• One new circular surface created by the cut (area = \(\pi r^2\))
• A curved surface (area = \(2\pi r h'\))

Total Surface Area of one new cylinder:

\( \text{TSA}_{\text{one new}} = \pi r^2 + \pi r^2 + 2\pi r h' = 2\pi r^2 + 2\pi r h' \)

Since there are two identical new cylinders, the total surface area of both pieces combined after cutting is:

\( \text{TSA}_{\text{after}} = 2 \times (\text{TSA}_{\text{one new}}) = 2 \times (2\pi r^2 + 2\pi r h') = 4\pi r^2 + 4\pi r h' \)

Substitute the values r = 28 cm and h' = 12 cm:

\( \text{TSA}_{\text{after}} = 4\pi (28 \text{ cm})^2 + 4\pi (28 \text{ cm})(12 \text{ cm}) \)

\( \text{TSA}_{\text{after}} = 4\pi (784 \text{ cm}^2) + 4\pi (336 \text{ cm}^2) \)

\( \text{TSA}_{\text{after}} = 3136\pi \text{ cm}^2 + 1344\pi \text{ cm}^2 \)

\( \text{TSA}_{\text{after}} = 4480\pi \text{ cm}^2 \)

4. Calculate the Increase in Total Surface Area

The increase in total surface area is the difference between the total surface area after cutting and the original total surface area.

\( \text{Increase} = \text{TSA}_{\text{after}} - \text{TSA}_{\text{original}} \)

\( \text{Increase} = 4480\pi \text{ cm}^2 - 2912\pi \text{ cm}^2 \)

\( \text{Increase} = 1568\pi \text{ cm}^2 \)

Alternatively, the increase comes from the two new surfaces created by the cut, each with area \(\pi r^2\). The total added area is \(2 \times \pi r^2 = 2\pi (28)^2 = 2\pi (784) = 1568\pi \text{ cm}^2\). This matches our calculation.

5. Calculate the Percentage Increase

The percentage increase is calculated using the formula:

\( \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{TSA}_{\text{original}}} \right) \times 100 \)

\( \text{Percentage Increase} = \left( \frac{1568\pi \text{ cm}^2}{2912\pi \text{ cm}^2} \right) \times 100 \)

Cancel out \(\pi\) and the units:

\( \text{Percentage Increase} = \left( \frac{1568}{2912} \right) \times 100 \)

Simplify the fraction:

\( \frac{1568}{2912} = \frac{1568 \div 16}{2912 \div 16} = \frac{98}{182} \)

\( \frac{98}{182} = \frac{98 \div 14}{182 \div 14} = \frac{7}{13} \)

So, the percentage increase is:

\( \text{Percentage Increase} = \left( \frac{7}{13} \right) \times 100 \)

\( \text{Percentage Increase} = \frac{700}{13} \)

Performing the division:

\( \frac{700}{13} \approx 53.846 \)

Rounding to two decimal places, the percentage increase is approximately 53.85 percent.

Revision Table: Cylinder Surface Area Calculation

Additional Information: Geometric Surface Area Changes

When a solid 3D object is cut, new surfaces are typically exposed, which increases the total surface area. The amount of increase depends on the shape of the object and the plane of the cut.

• For a cylinder cut horizontally, two new circles are formed at the cut plane. The area added is \(2 \times \pi r^2\).
• The curved surface area of the original cylinder (\(2\pi rh\)) is divided equally between the two new cylinders, so the curved surface area of each new cylinder is \(2\pi r(h/2) = \pi rh\).
• The base areas (\(\pi r^2\) for top and bottom) remain part of the new cylinders. Each new cylinder gets one original base.
• Total SA after cutting = (SA of piece 1) + (SA of piece 2)
• SA of one piece = (Original Base Area) + (New Cut Area) + (Curved Area of half height) = \(\pi r^2 + \pi r^2 + 2\pi r(h/2) = 2\pi r^2 + \pi r h\).
• Total SA after cutting = \(2 \times (2\pi r^2 + \pi r h) = 4\pi r^2 + 2\pi r h\). This matches the calculation \(4\pi r^2 + 4\pi r h'\) because \(h=2h'\).
• The increase in area is \((4\pi r^2 + 2\pi r h) - (2\pi r^2 + 2\pi r h) = 2\pi r^2\), which is the area of the two new surfaces. This confirms the approach.