Chapter 10: Problem 52
If the ratio of diagonals of two cubes is \(3: 2\) then the ratio of the surface areas of the two cubes respectively is : (a) \(5: 4\) (b) \(9: 5\) (c) \(9: 4\) (d) can't be determind
Short Answer
Expert verified
Answer: 9:4
Step by step solution
01
Finding side length from diagonal
Let's denote the side length of the first cube by \(a_1\) and the side length of the second cube by \(a_2\). According to the Pythagorean theorem, for a cube with side length \(a\), the diagonal is given by:
$$ d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} $$
For the two cubes, we have:
$$ d_1 = \sqrt{3a_1^2} $$
$$ d_2 = \sqrt{3a_2^2} $$
Now let's replace \(d_1\) and \(d_2\) with the given ratio.
02
Using the given ratio
We are given the ratio of the diagonals of the cubes to be \(3:2\). Hence:
$$ \frac{d_1}{d_2} = \frac{3}{2} $$
Now, replace \(d_1\) and \(d_2\) with the expressions in terms of side lengths:
$$ \frac{\sqrt{3a_1^2}}{\sqrt{3a_2^2}} = \frac{3}{2} $$
Now, solve for the ratio of the side lengths:
$$ \frac{a_1}{a_2} = \frac{3}{2} $$
03
Finding the surface areas
The surface area of a cube with side length \(a\) is given by the formula:
$$ A = 6a^2 $$
The surface areas of the cubes (\(A_1\) and \(A_2\)) are given by:
$$ A_1 = 6a_1^2 $$
$$ A_2 = 6a_2^2 $$
Now we need to find the ratio of surface areas.
04
Finding the ratio of surface areas
To find the ratio of the surface areas, we will divide \(A_1\) by \(A_2\):
$$ \frac{A_1}{A_2} = \frac{6a_1^2}{6a_2^2} $$
We can simplify this equation by canceling some terms:
$$ \frac{A_1}{A_2} = \frac{a_1^2}{a_2^2} $$
Now, substitute the given ratio of the side lengths and simplify:
$$ \frac{A_1}{A_2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$$
Hence, the ratio of the surface areas of the two cubes is \(9:4\). The correct answer is (c) \(9:4\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cube Geometry
In geometry, a cube is a three-dimensional structure that has six equal square faces, eight vertices, and twelve equal edges. Each angle between the edges is a right angle. This fascinating structure is beloved in geometry due to its symmetric and simple shape. Understanding cubes forms the basis for more complicated three-dimensional geometric concepts. Cubes have many interesting properties such as:
- The volume of a cube is given by the formula: \(V = a^3\), where \(a\) represents the side length of the cube.
- The surface area is calculated as \(A = 6a^2\), which is essentially the total area covered by all six square faces of the cube.
Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry, especially in dealing with right-angled triangles. It states that in such triangles, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it’s expressed as:\[c^2 = a^2 + b^2\]In the context of a cube, the theorem is used to find the cube diagonal by treating it as a right triangle’s hypotenuse where each side of the cube is one of the triangle’s legs. This is essential when calculating the length of a cube’s space diagonal, given as \(d = \sqrt{3a^2}\). Here, the diagonal runs from one corner of a cube to the opposite corner, spanning the interior of the cube.
Diagonals of a Cube
Diagonals of a cube are of two types:
- Face Diagonals: These lie on the square faces of the cube and can be calculated easily by applying the Pythagorean Theorem. For a face with side length \(a\), the face diagonal \(d_f\) is \(d_f = \sqrt{2}a\).
- Space Diagonals: These are the longer diagonals passing through the cube’s interior from one vertex to another opposite vertex. Using the Pythagorean Theorem, as shown previously, the space diagonal \(d\) is \(d = \sqrt{3}a\).
Ratio and Proportion
Ratio and proportion are mathematical concepts used to describe the relationship between quantities. A ratio is a way to compare two quantities, indicating how many times one value contains or is contained by the other. For example, a ratio of 3:2 means for every 3 units of the first quantity, there are 2 units of the second.
Proportions, on the other hand, help to find a missing quantity if the relation between a set of ratios is known. In the context of cubes, understanding ratios is key when deciphering problems involving comparative dimensions, such as diagonal or side length ratios, which in turn affect surface area and volume. For instance, once the ratio of the diagonals is known, the ratio of the side lengths can be inferred, and thus surface areas or volumes can be proportionately scaled.
By applying ratio and proportion, one can solve complex problems involving comparative scale, as they provide critical insight into how geometric changes in one dimension affect others.