Question

A family is using liquefied petroleum gas (LPG) of weight 14.2 kg for consumption. (Full weight 29.5 kg includes the empty cylinder's tare weight of 15.3 kg). If it is used with a constant rate then it lasts for 24 days. Then the new cylinder is replaced. Find the equation relating the quantity of gas in the cylinder to the days.

Answer 1

Hint: We will be using the concept of linear equations to solve the problem. We will then check the equation at end points to find the value of constants.

Complete step-by-step answer:
Now, we have been given that a family is using LPG of 14.2 kg for consumption. Also we have been given that the full weight 29.5 kg includes the empty cylinder.
Now, it has been given that the cylinder lasts for 24 days and we have to find an equation relating the quantity of gas in the cylinder to the days. Now, we have been given that the cylinder is used with constant rate. So, let the rate be c. Now, since the rate is constant we can express the relation between quantity of gas to the days with linear equation as,
\[Q\left( d \right)=cd+b.......\left( 1 \right)\]
Where b is any constant.
D is the nth day.
C is the rate of consumption.
Now, we know that at d = 0 the cylinder is full. Therefore, we have,
$\begin{align}
  & Q\left( 0 \right)=c\left( 0 \right)+b=14.2 \\
 & Q\left( 0 \right)=b=14.2 \\
\end{align}$
So, we have $b=14.2$.
Also, at d = 24. We have that the cylinder is empty. So,
$\begin{align}
  & Q\left( 24 \right)=c\left( 24 \right)+14.2=0 \\
 & c\left( 24 \right)=-14.2 \\
 & c=\dfrac{-14.2}{24} \\
\end{align}$
So, now we substitute the value of b and c in (1). So, that we have,
$Q\left( d \right)=\dfrac{-14.2}{24}\left( d \right)+14.2$

Note: To solve these types of questions it is important to note that the consumption of gas is at constant rate. Therefore, we have taken a general linear equation as,
\[Q\left( d \right)=cd+b\]