Question

The pressure of a certain quantity of a gas varies inversely as the volume. When the pressure is \[30\] units, volume is \[76\] units. Find the volume when pressure is \[30.4\] units.

Answer 1

Hint: In order to find the volume when pressure is \[30.4\] units, firstly we will be considering all the given measurements. And then we will be applying the proportionality theorem to the given quantities. Upon solving it accordingly, we will be obtaining the required answer.

Complete step by step answer:
Now let us briefly discuss proportions. . Proportion is nothing but saying that two ratios are equal. Two ratios can be written in proportion in the following ways- \[\dfrac{a}{b}=\dfrac{c}{d}\] or \[a:b=c:d\]. From the second way of notation, the values on the extreme end are called as extremes and the inner ones as means. Proportions are of two types: direct proportions and indirect or inverse proportions. In the direct proportion, there would be a direct relation between the quantities. In the case of indirect proportion, there exists an indirect relation between the quantities.
Now let us find the volume when pressure is \[30.4\] units.
So we are given that
\[{{P}_{1}}=30units\]
\[{{V}_{1}}=76units\]
\[{{P}_{2}}=30.4units\]
\[{{V}_{2}}=?\]
Now let us calculate the \[{{V}_{2}}\].
By applying the proportionality theorem and i.e. \[{{P}_{1}}{{V}_{1}}={{P}_{2}}{{V}_{2}}\]
Now let's substitute the values and solve it. We get
\[\begin{align}
  & {{P}_{1}}{{V}_{1}}={{P}_{2}}{{V}_{2}} \\
 & \Rightarrow 30\times 76=30.4\times {{V}_{2}} \\
 & \Rightarrow {{V}_{2}}=\dfrac{30\times 76}{30.4}=\dfrac{2280}{30.4}=75 \\
\end{align}\]
\[\therefore {{V}_{2}}=75units\]

Note: We must always assign the variable to the value to be found. In the above problem, the proportionality theorem we have applied is the same as Boyle’s law. We can also apply Boyle's law in solving this problem. We can apply proportions for finding the height of the buildings and trees and many more.