The distance of a port from 20 different locations form an AP.if the farthest distance is 300km, and the nearest distance is 30km, what is the distance between any two successive locations?
In this A.P. distances are:
a , a + r , a + 2 r ... , a + 19 r
where a is the distance of the nearest location from the port and r is the difference between any two successive location.
In this case a =30 km
The farthest distance is 300 km
The farthest distance is 20th term of A.P.
a20 = a + 19 r
a + 19 r = 300
30 + 19 r = 300
19 r = 300 - 30
r = 270 /19 km
270 / 19 km is the distance between any two successive location.
Well, if the nearest distance is 30km and the farthest distance is 300km, I'm afraid it seems like quite a stretch. But fear not! The distance between any two successive locations can be found by simply dividing the total range of distances (300km - 30km = 270km) by the number of locations minus one, since there are 20 locations in total. So, let me grab my little calculator here... *beep boop boop* Ah, yes. The distance between any two successive locations in this case would be 270km divided by 19, which is approximately 14.21km. So, get ready to pack your bags and enjoy those little jumps from one location to another!
To find the distance between any two successive locations, we need to determine the common difference of the arithmetic progression (AP) formed by the distances of the port from the 20 different locations.
The first term (a) of the AP is the nearest distance from the port, which is 30 km.
The nth term (an) of the AP is the farthest distance from the port, which is 300 km.
The number of terms in the AP (n) is 20.
We can use the formula for the nth term of an AP to find the common difference (d):
an = a + (n - 1)d
Substituting the known values:
300 = 30 + (20 - 1)d
300 = 30 + 19d
19d = 300 - 30
19d = 270
d = 270 / 19
d ≈ 14.21
Therefore, the distance between any two successive locations is approximately 14.21 km.
To find the common difference between any two successive locations in an arithmetic progression (AP), we can use the formula:
Common Difference (d) = (Farthest Distance - Nearest Distance) / (Number of Terms - 1)
In this case, the farthest distance is 300 km, the nearest distance is 30 km, and the number of terms is 20.
Substituting these values into the formula:
d = (300 - 30) / (20 - 1)
d = 270 / 19
d ≈ 14.21 km (rounded to two decimal places)
Therefore, the distance between any two successive locations in this scenario is approximately 14.21 km.