Question

This is a map of three towns. It shows the train line and the highway that connects the towns. The north direction is shown. The map is drawn to scale. The distance between Tounouse and Avaria is 250 kilometers. A new train line is to be built between Tounouse and Mt.Monotti. Estimate the distance between these two towns.

Answer 1

Answer:

  about 106 km

Step-by-step explanation:

The distance between the two towns can be estimated several ways. In the attachment, we chose to measure the angles on the map. Using the law of sines, we can estimate the unknown distance.

The distance can also be estimated using the Tounouse-Avaria distance as a ruler. For best results, that ruler would need to be subdivided to an appropriate level.

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angle measures

The angles in the triangle can be measured using a protractor or some other tool. The attachment shows the results from a geometry program:

• ∠TMA ≈ 107°
• ∠TAM ≈ 24°

law of sines

The law of sines tells us the distances are proportional to the sines of the opposite angles:

  TM/sin(TAM) = TA/sin(TMA)

  TM = sin(24°)×(250 km)/sin(107°) ≈ 106.3 km

The distance from Tounouse to Mt. Monotti is estimated to be about 106 km.

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Alternate solution

Using a compass or dividers to project the TM distance onto TA, we find it is between 0.4 and 0.5 of the TA distance. The difference between 2×TM and TA can be estimated to be about 0.4×TM. That is, ...

  TM ≈ TA/2.4 ≈ (250 km)/2.4 ≈ 104.2 km

This estimate is consistent with the one found using angle measures.