Streetlamps are positioned at intervals of 240 m along a road and angsana trees are planted at intervals of 192 m. The first angsana tree is planted beside the first streetlamp.
(a)If the last angsana tree if planted beside the last tree lamp, find the least possible distance between the first and the last angsana tree.
(b)Find, using your answer in (a), the total number of streetlamps and angsana trees within this section of the road.
To find the least possible distance between the first and last angsana tree, we need to find the least common multiple (LCM) of 240 and 192.
We can find the LCM by finding the prime factorization of both numbers:
240 = 2^4 * 3 * 5
192 = 2^6 * 3
To find the LCM, we take the highest power of each prime factor that appears in the factorizations:
LCM = 2^6 * 3 * 5 = 960
So the least possible distance between the first and last angsana tree is 960 meters.
To find the total number of streetlamps and angsana trees within this section of the road, we need to find the number of intervals of 240 and 192 that fit within a distance of 960 meters.
First, let's find the number of intervals of 240:
960 / 240 = 4
There are 4 intervals of 240, so there are 4 streetlamps.
Next, let's find the number of intervals of 192:
960 / 192 = 5
There are 5 intervals of 192, so there are 5 angsana trees.
Therefore, the total number of streetlamps and angsana trees within this section of the road is 4 streetlamps and 5 angsana trees.