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4.2 B) Solving Loci Problems
4.2 B) Solving Loci Problems
We are going to be looking at a problem solving loci question in this section. Before you work through this section, make sure that you have covered the content in the previous section (click here to be taken back to the previous section).
Click here for a printable version of the question in this section.
Click here for a printable version of the question in this section.
Example 1
There are two telephone masks on the diagram below; mask A and mask B. The scale of the diagram is 1 cm = 2 km.
There are two telephone masks on the diagram below; mask A and mask B. The scale of the diagram is 1 cm = 2 km.
Mask A has a range of 8 km and mask B has a range of 10 km. Find on the diagram the area that is able to get a signal from both of the telephone masks.
Both of the telephone masks are points and this means that the loci will be a circle (the method used to find the loci will be the same as the method used in the horse example in the previous section). All of the points inside the circular loci will be able to receive signal.
I am going to obtain the locus for mask A first. Before we are able to draw the locus on the diagram, we need to find the range of transmitter A on the diagram. We find the range of transmitter A on the diagram by dividing the actual range of the transmitters by the scale. We are told in the question that the range of transmitter A is 8 km and the scale on the diagram is 1 cm is 2 km. Therefore, we find the range of the transmitter on the diagram by completing the following calculation:
Both of the telephone masks are points and this means that the loci will be a circle (the method used to find the loci will be the same as the method used in the horse example in the previous section). All of the points inside the circular loci will be able to receive signal.
I am going to obtain the locus for mask A first. Before we are able to draw the locus on the diagram, we need to find the range of transmitter A on the diagram. We find the range of transmitter A on the diagram by dividing the actual range of the transmitters by the scale. We are told in the question that the range of transmitter A is 8 km and the scale on the diagram is 1 cm is 2 km. Therefore, we find the range of the transmitter on the diagram by completing the following calculation:
The range of transmitter A on the diagram is 4 cm. We are now ready to draw the locus. The locus for the range will be circular. We draw a circular locus by opening the compass up so that the distance between the point and the pencil on the compass is 4 cm. We then put the point of the compass on transmitter A and spin the compass around creating a locus. The outcome is shown below:
We do the same for transmitter B. Like with mask A, the first step is to find out the range of transmitter B on the diagram, which we are able to do by dividing the actual range by the scale. We are told in the question that the range of transmitter B is 10 km and the scale is 1 cm is 2 km. Therefore, we find the range on the diagram by completing the calculation below:
The range of transmitter B on the diagram is 5 cm. The next step is to measure 5 cm out on a compass, place the point of the compass on transmitter B and spin the compass around. The outcome is shown on the diagram below:
We now have the ranges of both of the transmitters. The question asks us to find the area on the diagram that gets signal from both of the transmitters. This area will be where the two loci circles overlap one another. I have labelled this area on the diagram below.