$1764\pi \text{ cm}^2$
Sphere Surface Area Calculation
This solution explains how to find the surface area of a sphere when the ratio of its volume to surface area is given.
Understanding the Formulas
- Volume of a sphere ($V$): $V = \frac{4}{3}\pi r^3$
- Surface area of a sphere ($A$): $A = 4\pi r^2$
- Here, '$r$' represents the radius of the sphere.
Calculating the Radius
The problem states that the ratio of the volume to the surface area is 7 cm:
$ \frac{V}{A} = 7 \text{ cm} $
Substitute the formulas for volume and surface area:
$ \frac{\frac{4}{3}\pi r^3}{4\pi r^2} = 7 $
Simplify the expression:
$ \frac{4\pi r^3}{3 \times 4\pi r^2} = 7 $
Cancel out common terms ($4\pi r^2$):
$ \frac{r}{3} = 7 $
Solve for the radius ($r$):
$ r = 7 \times 3 $
$ r = 21 \text{ cm} $
Finding the Surface Area
Now, use the calculated radius ($r = 21$ cm) to find the surface area ($A$) using its formula:
$ A = 4\pi r^2 $
Substitute the value of $r$:
$ A = 4\pi (21)^2 $
Calculate $21^2$:
$ A = 4\pi (441) $
Multiply to find the final surface area:
$ A = 1764\pi \text{ cm}^2 $
The surface area of the sphere is $1764\pi \text{ cm}^2$.
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.)
