\(6(x+y)\)
To solve the problem, we need to find the total surface area of a cube given the sum of the lengths of a diagonal of a face of the cube and a diagonal of the cube. Let's break the problem down step-by-step:
- Let \(a\) be the side length of the cube. The diagonal of a face of the cube is the diagonal of a square with side \(a\). Therefore, the diagonal \(x\) of a face is given by: \(x = \sqrt{a^2 + a^2} = \sqrt{2}a\).
- The diagonal \(y\) of the cube itself can be found using the Pythagorean theorem in three dimensions: \(y = \sqrt{a^2 + a^2 + a^2} = \sqrt{3}a\).
- According to the problem, \(x + y = 5 + 2\sqrt{6}\). Plugging in the expressions for \(x\) and \(y\), we have: \(\sqrt{2}a + \sqrt{3}a = 5 + 2\sqrt{6}\).
- Simplifying this, we get: \(a(\sqrt{2} + \sqrt{3}) = 5 + 2\sqrt{6}\).
- We need the total surface area of the cube, which is given by \(6a^2\).
- Given the options, instead of finding \(a\), we look at the option \(6(x+y)\). This is consistent with obtaining the total surface area when related variables \(x\) and \(y\) are summed and multiplied by 6: \(6(x + y) = 6(5 + 2\sqrt{6})\).
- Thus, the correct answer is \(6(x+y)\), since this matches directly as the computation required for the total surface area instead of duplicating steps to find and verify individual \(a^2\).
Therefore, the total surface area of the cube is given by the expression \(6(x+y)\). The correct answer is \(6(x+y)\).
Similar Questions
- A sphere of diameter 14 cm is dropped into a cylindrical vessel partly filled with water. The radius of the vessel is 14 cm. If the sphere is completely submerged in water, then how much will the level of water rise ?
- The diameter of a copper solid sphere is 6 cm. The sphere is melted and recast into a wire. If the diameter of the wire is 0.5 cm, then what is the length of the wire ?
- A metallic sheet of rectangular shape is 48 cm × 36 cm. From each corner a square of side x cm is cut off and an open box is made of the remaining sheet. If the box holds 5.12 litre of water, then what is the value of x ?
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 = \(22\over7\)]
Which of the following is a geometrical figure with a three-dimensional geometry that has eight vertices and six rectangular faces?
- 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 $\pi = \frac{22}{7}$
