Sphere Surface Area Increase: Calculating the Percentage Change

This solution explains how to calculate the percentage increase in the total surface area of a solid metallic sphere when it is cut into 8 identical pieces using three mutually perpendicular cuts passing through its center.

Understanding Original Sphere Surface Area

First, let's consider the original solid metallic sphere. Let the radius of this sphere be denoted by r.

The formula for the surface area of a sphere is:

$A_{\text{original}} = 4\pi r^2$

This represents the total area covering the outside of the sphere.

Analyzing Perpendicular Cuts Effect

The problem states the sphere is cut into 8 identical pieces using 3 mutually perpendicular cuts that go through the sphere's center. This method effectively divides the sphere into 8 equal parts, like slicing an orange into 8 identical wedges.

Each cut introduces new surfaces that were previously in the interior of the sphere. Since each cut is a plane passing through the center, it creates a flat circular surface, which is a 'great circle' of the sphere. The area of one such great circle is:

$A_{\text{cut}} = \pi r^2$

Each cut creates two new flat surfaces (one on each side of the cut plane). Therefore, the total new surface area exposed depends on the number of cuts made.

Calculating Total Surface Area of Pieces

We are making 3 mutually perpendicular cuts. The total new surface area exposed inside the sphere is calculated as follows:

Total new area created = (Number of cuts) × (Number of new surfaces per cut) × (Area of one cut surface)

Total new area created = $3 \times 2 \times (\pi r^2) = 6\pi r^2$.

The total surface area of the 8 resulting pieces, denoted as $A_{\text{total\_8}}$, is the sum of the original external surface area and the total newly exposed internal surface area:

$A_{\text{total\_8}} = A_{\text{original}} + (\text{Total new area created})$

Substituting the values we have:

$A_{\text{total\_8}} = 4\pi r^2 + 6\pi r^2$

$A_{\text{total\_8}} = 10\pi r^2$

Determining Surface Area Percentage Increase

To find the percentage increase, we first calculate the absolute increase in surface area:

Increase in Area = $A_{\text{total\_8}} - A_{\text{original}}$

Increase in Area = $10\pi r^2 - 4\pi r^2 = 6\pi r^2$.

Now, we calculate the percentage increase relative to the original surface area:

$\text{Percentage Increase} = \frac{\text{Increase in Area}}{A_{\text{original}}} \times 100\%$

Plugging in the values:

$\text{Percentage Increase} = \frac{6\pi r^2}{4\pi r^2} \times 100\%$

The term $\pi r^2$ cancels out from the numerator and denominator:

$\text{Percentage Increase} = \frac{6}{4} \times 100\%$

$\text{Percentage Increase} = \frac{3}{2} \times 100\%$

$\text{Percentage Increase} = 1.5 \times 100\% = 150\%$

Thus, the total surface area of the 8 pieces is 150% greater than the surface area of the original sphere.