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)
- The height of a cylinder is 14cm and its curved surface area is 264cm². The volume of the cyclinder (in cm³) is:
($\pi=\frac{22}{7}$) - A cylindrical rod has an outer curved surface area of \(7500 \text{ cm}^2\). If the length of the rod is 92 cm, then the outer radius (in cm) of the rod, rounded off to two places of decimal, is:
\(\left(\text{Take } \pi = \frac{22}{7}\right)\) - A number of 512 identical small spheres are cast from a sphere of radius 40 cm, with the total volume of the small spheres being equal to the volume of the larger sphere. The diameter (in cm) of each of the small spheres is:
There is a wooden block in the form of a cube whose each side is 8 meters long.
The maximum possible number of cylinders with a diameter of 1 meter and a height of 4 meters were cut from this block. The cylinders are to be painted at the rate of ₹14 per square meter.
What is the total amount (in ₹) needed to paint all the cylinders if we paint the entire surface of each cylinder? (Take $\pi = \frac{22}{7}$)
- If the lateral surface area of a cylinder is 775.8 cm$^2$ and its height is 24 cm, then find its volume. (Use $\pi = 3.14$ and round off to two decimal places.)
