Answer

Verified

429.3k+ views

**Hint:**We solve this problem by considering the height of cistern as \['H'\] and the time taken by the leakage to empty it as \['x'\] then we use the given condition that when both pipes are open it takes 30 minutes extra to fill the tank due to leakage to find the required value. While applying the conditions we need to keep in mind that times of individual taps should not be added to get time for filling tanks together. We need to add the height not time.

**Complete step-by-step answer:**

Let us assume that the height of cistern as \['H'\]

We are given that the first pipe takes 12 hours to fill the tank.

So, we can write that the first pipe fills \['\dfrac{H}{12}'\] height of the cistern in 1 hour.

We are given that the second pipe takes 20 hours to fill the tank.

So, we can write that the second pipe fills \['\dfrac{H}{20}'\] height of the tank in 1 hour.

Let us assume that the height of tank filled with both the pipes simultaneously as \['h'\] then we get

\[\Rightarrow h=\dfrac{H}{12}+\dfrac{H}{20}\]

Let us assume that both pipes together take \['t'\] time to fill the tank without leakage then we get

\[\begin{align}

& \Rightarrow t=\dfrac{H}{\dfrac{H}{12}+\dfrac{H}{20}} \\

& \Rightarrow t=\dfrac{12\times 20}{12+20} \\

& \Rightarrow t=\dfrac{15}{2}hrs \\

\end{align}\]

Here, we can say that it takes \[\dfrac{15}{2}\] hours to fill the tank by both pipes together.

Let us assume that the time taken to fill the tank when there is leakage to empty the tank as \['x'\]

We are given that when both pipes are open it takes 30 minutes extra to fill the tank due to leakage

We know that the conversion of time that is

\[1hr=60\min \]

By using this conversion we get

\[\Rightarrow 30\min =\dfrac{1}{2}hr\]

Now, by converting the given condition into mathematical equation we get

\[\Rightarrow t+\dfrac{1}{2}=x\]

Now, by substituting the value of \['t'\] in above equation we get

\[\begin{align}

& \Rightarrow x=\dfrac{15}{2}+\dfrac{1}{2} \\

& \Rightarrow x=8hr \\

\end{align}\]

Therefore we can say that it takes 8 hours to fill the tank when there is leakage.

So, we can say that both pipes combined with leakage empties \['\dfrac{H}{8}'\] height of the tank in 1 hour.

Let us assume that the leakage takes \['y'\] hours to empty the tank.

So, we can say that the leakage empties \['\dfrac{H}{y}'\] height in one hour.

Now, we can say that the height filled by both pipes to fill the tank when there is leakage as

\[\begin{align}

& \Rightarrow \dfrac{H}{12}+\dfrac{H}{20}-\dfrac{H}{y}=\dfrac{H}{8} \\

& \Rightarrow \dfrac{1}{y}=\dfrac{1}{12}+\dfrac{1}{20}-\dfrac{1}{8} \\

\end{align}\]

Now by adding the terms using the LCM we get

\[\begin{align}

& \Rightarrow \dfrac{1}{y}=\dfrac{10+6-15}{120} \\

& \Rightarrow y=120 \\

\end{align}\]

Therefore, we can say that the leakage takes 120 hours to empty the tank.

**Note:**Students may make mistakes in taking the time of both pipes together to fill the tank.

We have the time taken by both pipes to fill the tank as

\[\Rightarrow t=\dfrac{H}{\dfrac{H}{12}+\dfrac{H}{20}}\]

Bu students may do mistake and add the times directly to get the time taken by both pipes together that is

\[\Rightarrow t=12+20\]

This gives the wrong answer because we are not allowed to add the times when the situations are taken together. But instead, we can add work done in small change in time as we considered 1 hour in the solution.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths