Cube Volume Calculation from Painting Cost
The problem asks for the volume of a cube, given the total cost of painting all its external surfaces and the painting rate.
Step 1: Calculate the Total Surface Area (TSA)
First, find the total surface area that was painted. This is done by dividing the total cost by the rate per square centimeter.
- Total Cost = ₹588
- Rate = ₹2/cm$^2$
-
Total Surface Area (TSA) = $\frac{\text{Total Cost}}{\text{Rate}}$
TSA = $\frac{₹588}{₹2/\text{cm}^2}$ = 294 cm$^2$
Step 2: Determine the Side Length of the Cube
The formula for the total surface area of a cube with side length '$a$' is $6a^2$. We use the TSA calculated in Step 1 to find '$a$'.
- TSA = $6a^2$
- $294 \text{ cm}^2 = 6a^2$
- $a^2 = \frac{294 \text{ cm}^2}{6}$
- $a^2 = 49 \text{ cm}^2$
- $a = \sqrt{49 \text{ cm}^2}$
- $a = 7 \text{ cm}$
Step 3: Calculate the Volume of the Cube
Finally, calculate the volume of the cube using the side length '$a$' found in Step 2. The formula for the volume (V) of a cube is $a^3$.
- Volume (V) = $a^3$
- V = $(7 \text{ cm})^3$
- V = $7 \times 7 \times 7$ cm$^3$
- V = 343 cm$^3$
The volume of the cube is 343 cm$^3$. This matches option B.
Similar Questions
- The radius of a sphere is 16 cm. Find its surface area.
- What is the ratio of volumes of two spheres where the curved surface areas are in the ratio of $1 : 4$?
- Five solid cubes, each of volume $39304 \text{ cm}^3$, are joined end to end to form a cuboid. What is the lateral surface area (in $\text{cm}^2$) of the cuboid?
- What is the surface area of a sphere of radius 8 cm? (use $\pi = 3.14$)
- The ratio of the volumes of two cubes is 64 : 1331. What is the ratio of their total surface areas?
- The diameter of a hemisphere is 7 cm. What will its total surface area be?
- The diameters of the bases of two cones are equal. If their slant heights are in the ratio of 3 : 4, then what will the ratio of their curved surface areas be?
- A cube of volume 324 cm$^3$ is molded into a cuboid whose length, width and height are in the ratio $3 : 2 : 2$. Find the length of the cuboid.
- An aquarium is in the form of a cuboid whose external measures are $80 \text{ cm} \times 30 \text{ cm} \times 40 \text{ cm}$. The base, side faces and back face are to be covered with a coloured paper. Find the area of the paper needed?
- A godown is in the form of a cuboid of measures $60 \text{ m} \times 40 \text{ m} \times 30 \text{ m}$. How many cuboidal boxes can be stored in it if the volume of one box is $0.8 \text{ m}^3$?
Important Questions from 3-D Mensuration
Find the total surface area of a closed cylinder having a base radius of 70 m and a height of 110 m. [Use π = \(22\over7\)]
- The curved surface area of a cylinder is half of its total surface area. If its height is 195 cm, then find its diameter (in cm).
The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone.
(Use $\pi = \frac{22}{7}$)- A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$ - The volume (in $m^3$) of a cube, each of whose edges is 42 m, is:
