Question

Write the following statements using symbols of variation.
(1) The wavelength of sound (l) and its frequency (f) are in inverse variation.
(2) The intensity of light varies inversely with the square of the distance (d) of a screen from the lamp.

Answer 1

Hint: We will start solving this question by using the theory of proportions. There are two types of proportion – direct and inverse proportion.

Complete step-by-step solution -
Now, according to the theory of proportion, we can establish a relation between any two quantities. Establishing a relation between various quantities helps us in day to day life. With the help of proportions, every formula that we use in Physics, Chemistry are derived. There are two types of proportions,
A). Direct proportion
B). Indirect proportion
Direct proportion is used where there is a direct relation between quantities. For example, how much you study is directly dependent upon how much you understand. There is a direct relation.
Indirect proportion is used where there is an indirect relation between quantities. For example, how much you study is indirectly dependent upon how much time you waste. If you waste less time, you can study more and vice-versa.
Now, proportions are represented by using the symbol $\alpha $. For direct proportionality, we can write a $\alpha $b, where a is directly proportional to b. We can also write it as, a = kb, where k is a constant.
For indirect proportionality, we can write as, a $\alpha $ $\dfrac{1}{b}$, where a and b are indirectly proportional to each other.
Now, the wavelength of sound (l) is indirectly proportional to frequency (f) as they are in inverse variation. So, we can write, as
l $\alpha $ $\dfrac{1}{f}$
or we can write $l = \dfrac{k}{f}$ , where k is a constant.
Now, we have intensity of light (I) is inversely proportional to square of distance (d) from screen, so
I $\alpha $ $\dfrac{1}{{{d^2}}}$,
or we can write it as, $I = \dfrac{A}{{{d^2}}}$, where A is another constant.

Note: Whenever we come up with such types of questions, we will first read the question. If it is written that the given quantities are in inverse variation to each other, then we will make a relation between given quantities by placing a symbol $\alpha $ between them and taking the reciprocal to one of the quantities. For example, if a and b are in inverse variation, so we write a $\alpha $ $\dfrac{1}{b}$. Here, we have taken the reciprocal to b and use the symbol$\alpha $. If direct variation is written, then we will place$\alpha $in between the quantities, for example, a $\alpha $b, representing a is directly proportional to b.