Question

A spherical steel ball was silver polished then it was cut into 4 similar pieces. What is ratio of the polished area to the non polished area: (a) \(1: 1\) (b) \(1: 2\) (c) \(2: 1\) (d) can't be determined

Short answer

Answer: The ratio of the polished area to the non-polished area is 1:2.

Step-by-step solution

Calculate the total surface area of the sphere

Let the radius of the sphere be \(r\). The total surface area of the sphere can be found using the formula: \(A_\text{sphere} = 4 \pi r^2.\)

Calculate the surface area of a single piece

Since the sphere is cut into 4 similar pieces, we can find the surface area of one piece, then multiply by 4 to get the total surface area. Each piece can be regarded as a hemisphere with curved surface area plus the area of a circle(base) where it is cut. So, the surface area of a single piece is: \(A_\text{piece} = 2\pi r^2 + \pi r^2 = 3\pi r^2.\)

Calculate the total polished area

The total polished area is the surface area of the original sphere. Therefore, \(A_\text{polished} = A_\text{sphere} = 4\pi r^2.\)

Calculate the total non-polished area

The total non-polished area can be found by subtracting the polished area from the total surface area of the four pieces. We multiply the area of one piece by 4 and then subtract the polished area: \(A_\text{non-polished} = 4A_\text{piece} - A_\text{polished} = 4(3\pi r^2) - 4\pi r^2 = 8\pi r^2.\)

Find the ratio of polished to non-polished area

Now, we divide the polished area by the non-polished area to find the ratio: \(A_\text{ratio} = \frac{A_\text{polished}}{A_\text{non-polished}} = \frac{4\pi r^2}{8\pi r^2} = \frac{1}{2}.\) Converting this fraction to a ratio, we get the ratio of polished area to non-polished area as \(1: 2\). So, the correct answer is (b) \(1: 2\).

Key concepts

Geometry of a Sphere

In geometry, a sphere is a perfectly round three-dimensional shape, similar to a ball. Understanding the surface area of a sphere is important for tackling many geometry problems. The surface area of a sphere is given by the formula: \( A_\text{sphere} = 4 \pi r^2 \), where \( r \) is the radius of the sphere. This formula helps determine the total area that covers the entire outer surface of a sphere.
When a sphere is cut into smaller parts, as in our problem, each piece retains a portion of this surface area. In this case, the sphere is divided into four equal parts. Each part resembles a hemisphere with an additional circular base. It's crucial in geometry to think about the shapes resulting from such divisions, as it affects how you approach surface area calculations.
This type of geometric thinking is vital as it allows you to predict how changes in physical properties like curvature or size might affect surface area calculations.

Problem Solving in Mathematics

Mathematics problem solving involves breaking down a problem into manageable steps. Our given exercise does just that by first calculating the total surface area of the sphere. Then, we compute the surface area for each of the smaller pieces individually.

• The first step is to understand the problem, in this case, finding areas that are polished versus non-polished.
• Next, you apply relevant formulas, starting with the surface area of the whole sphere.
• Then, determine the surface area of each smaller part and multiply by the number of parts. This gives you the total surface area of those parts.
• Finally, compare these areas to get the desired ratio.

Breaking down problems into steps makes them easier to solve and reduces errors, which is key in any math-related task.

Understanding Ratios

Ratios are a way to express the relationship between two quantities. In our problem, we are asked to compare the polished area and non-polished area of the sphere when divided into four pieces.
To calculate the ratio, you divide one quantity by the other. From our previous calculation steps, we find the polished area to be \( 4 \pi r^2 \) and the non-polished area to be \( 8 \pi r^2 \).
By dividing the two, we derive the ratio \( \frac{4\pi r^2}{8\pi r^2} \), which simplifies to \( \frac{1}{2} \). This results in the ratio of 1:2, indicating that for every unit of polished area, there are two units of non-polished area.
This understanding of ratios is fundamental in mathematics as it helps in comparing quantities and understanding their proportions.