Spherical Balloon Expansion: Color Patch Area Calculation
This problem involves understanding how the area of a feature on a spherical surface changes when the sphere itself expands uniformly. We are given the initial and final radii of a spherical balloon and the initial area of a color patch on its surface. We need to find the new area of the color patch after the balloon expands.
Understanding Surface Area Scaling
When a sphere expands uniformly, its surface area scales with the square of its radius ($r^2$). Since the color patch is a part of the balloon's surface, its area will also scale proportionally. Assuming the patch maintains its relative size and shape on the surface as the balloon expands, its area ($A$) is directly proportional to the square of the radius.
We can express this relationship mathematically:
$$ A \propto r^2 $$
This implies that the ratio of the areas ($A$) of the color patch at two different stages (initial state 1, final state 2) is equal to the square of the ratio of their corresponding radii ($r$):
$$ \frac{A_2}{A_1} = \left(\frac{r_2}{r_1}\right)^2 $$
Where:
- $A_1$ = Initial area of the color patch
- $r_1$ = Initial radius of the balloon
- $A_2$ = Final area of the color patch
- $r_2$ = Final radius of the balloon
Step-by-Step Calculation
Let's apply the derived formula using the values provided in the question:
- Initial radius, $r_1 = 10$ cm
- Initial area, $A_1 = 25$ cm²
- Final radius, $r_2 = 50$ cm
We need to determine the final area, $A_2$.
Substitute the known values into the scaling formula:
$$ \frac{A_2}{25 \text{ cm}^2} = \left(\frac{50 \text{ cm}}{10 \text{ cm}}\right)^2 $$
First, calculate the ratio of the final radius to the initial radius:
$$ \frac{r_2}{r_1} = \frac{50}{10} = 5 $$
Next, square this ratio:
$$ \left(5\right)^2 = 25 $$
Now, the equation simplifies to:
$$ \frac{A_2}{25 \text{ cm}^2} = 25 $$
To find the final area ($A_2$), multiply both sides of the equation by 25 cm²:
$$ A_2 = 25 \times 25 \text{ cm}^2 $$
$$ A_2 = 625 \text{ cm}^2 $$
Therefore, the area of the color patch on the expanded spherical balloon is 625 cm².
Important Questions from Mensuration 3D (Notes)
- A block of marble 5 m x 4 m x 2 m in size is cut into rectangular tiles of 1 m x 0.5 m size having thickness of 10 cm. Assuming 10% wastage in cutting, how many tiles will be made?
- The height of a cylinder is 14cm and its curved surface area is 264cm². The volume of the cyclinder (in cm³) is:
($\pi=\frac{22}{7}$) - What is the volume of a 6 m deep tank having rectangular shaped top 6m X 4 m and bottom 4 m X 2 m? (use mean-area method).
- The surface area of the solid generated by revolving the curve $x = e^t \cos t, y = e^t \sin t$ about y-axis $0 \leq t \leq \pi/2$ is
- The surface area of the plane $x + 2y + 2z = 12$ cut off by $x = 0, y = 0$ and $x^2 + y^2 = 16$ is
