Draining a pool

Answer: 8 minutes

Using similar triangles

 

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Triangles are similar so 21 : 3 = 56 : ?

21 : 3 = 7 : 1

56 = 7$\times$8 so 56 : 8 is also the same as 7 : 1, ? = 8.

Using rates of change

In the first 3 minutes, the depth of the water decreases by 21 cm.

So in each 1 minute, depth decreases by 7 cm.

To reach 144 cm, it must decrease by 35 cm more, since 144 + 35 = 179.

35 = 7$\times$5, so it will take 5 more minutes, or 8 minutes in total.

Using gradients

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In $3$ minutes, the depth of the water goes down by $21$ cm.

So the gradient of the line is $\frac{-21}3=-7$.

When the water level is $144$ cm, the depth of the water will have gone down by $56$ cm.

So, if this happens at time $t$, $\frac{-56}t=-7$, so $t=8$.

Using proportion

time	decrease	final depth
3 minutes	21 cm	179 cm
3 more min	21 cm	158 cm
1 more min	7 cm	151 cm
1 more min	7 cm	144 cm

Total 8 minutes.

Using congruent triangles

Because the graph is a straight line, we can use congruent triangles congruent to find other points on the graph, as shown on the right.

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These three triangles have taken us too far, so we can now use smaller similar triangles to find more information:

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We can see that after 8 minutes, the height of the water will be 144 cm.