Understanding the Moon-Earth Diameter Ratio
The problem states that the diameter of the Moon is approximately one-fourth the diameter of the Earth. We need to find the ratio of their surface areas based on this information.
Relating Diameter, Radius, and Surface Area
To find the ratio of the surface areas, we first need to understand how diameter, radius, and surface area are related.
- Diameter and Radius: The radius ($R$) of a sphere is half its diameter ($D$). So, $R = \frac{D}{2}$.
- Surface Area Formula: The surface area ($A$) of a sphere is calculated using the formula $A = 4 \pi R^2$.
Step-by-Step Calculation
Let's denote the diameter and radius of the Moon as $D_M$ and $R_M$, and those of the Earth as $D_E$ and $R_E$.
- Diameter Relationship: We are given that $D_M = \frac{1}{4} D_E$.
- Radius Calculation:
- The radius of the Moon is $R_M = \frac{D_M}{2}$. Substituting the given diameter relationship, we get $R_M = \frac{(\frac{1}{4} D_E)}{2} = \frac{1}{4} \left( \frac{D_E}{2} \right)$.
- Since $R_E = \frac{D_E}{2}$, we can write the Moon's radius in terms of the Earth's radius: $R_M = \frac{1}{4} R_E$. This means the Moon's radius is also one-fourth the Earth's radius.
- Surface Area Ratio:
- The surface area of the Moon is $A_M = 4 \pi R_M^2$.
- The surface area of the Earth is $A_E = 4 \pi R_E^2$.
- The ratio of their surface areas is $\frac{A_M}{A_E}$.
- Substituting Radii: We can substitute the relationship $R_M = \frac{1}{4} R_E$ into the ratio formula: $ \frac{A_M}{A_E} = \frac{4 \pi R_M^2}{4 \pi R_E^2} = \frac{R_M^2}{R_E^2} = \left( \frac{R_M}{R_E} \right)^2 $ $ \frac{A_M}{A_E} = \left( \frac{\frac{1}{4} R_E}{R_E} \right)^2 = \left( \frac{1}{4} \right)^2 $
- Final Ratio: Calculating the square: $ \left( \frac{1}{4} \right)^2 = \frac{1^2}{4^2} = \frac{1}{16} $
Conclusion on Surface Area Ratio
The ratio of the surface area of the Moon to the surface area of the Earth is 1 : 16.
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 $140.1 \text{ cm}^2$ and its height is $3 \text{ cm}$, then find its volume. (Use $\pi = 3.14$ and round off to two decimal places.)
