# Draining a pool

The water is being drained from a pool. After how long will the depth of the pool be 144 cm?

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The graph shows the depth of the water, in centimetres, against time, in minutes.

At what time will the depth of the water be 144 cm?

*This problem is taken from the World Mathematics Championships*

**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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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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