SOLUTION: (a) The eighth term of an Arithmetic Progression (A.P) is 46 and the sum of the first eight terms is 200. Find the: (i) first term; (ii) sum of the first 12 terms. (b) The

Algebra ->  Test -> SOLUTION: (a) The eighth term of an Arithmetic Progression (A.P) is 46 and the sum of the first eight terms is 200. Find the: (i) first term; (ii) sum of the first 12 terms. (b) The      Log On


   

Question 1210336: (a) The eighth term of an Arithmetic Progression (A.P) is 46 and the sum of the first eight terms is 200. Find the:
(i) first term;
(ii) sum of the first 12 terms.

(b) The points X(70°S, 60°E) and Y(7°S, 60°E) lies on the surface of the earth.
(i) Illustrate the information in a diagram.
(ii) Find the distance between X and Y along the meridian.
[Take л = 22/7 and R = 6,400 km]
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


(a)

The 8th term is 46; the sum of the first 8 terms is 200.

The sum of the first 8 terms is the number of terms (8) times the average of all the terms; the average of all the terms is the average of the first and last. So to find the first term a:

%28a%2B46%29%2F2=200%2F8=25
a%2B46=50
a=4

The 1st term is 4; the 8th term, 46, is the first term plus the common difference d 8-1 = 7 times. The common difference d is

d=%2846-4%29%2F%288-1%29=42%2F7=6

The 12th term is the 8th term, 46, plus the common difference 12-8 = 4 times:

46%2B4%286%29=46%2B24=70

The sum of the first 12 terms is the number of terms, 12, times the average of the first and 12th terms:

12%28%284%2B70%29%2F2%29=12%2837%29=444

ANSWERS: first term 4; sum of first 12 terms 444

(b)

The difference between the two points is 70-7 = 63 degrees in latitude.

Latitudes range from -90 degrees (South Pole) to +90 degrees (North Pole), a range of 180 degrees.

The distance between the two points is (63/180)=(7/20) of the distance from the South to North Poles, which is (7/20) of the length of a semicircular arc with radius 6400 km.



ANSWER: 3520 km

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
(b) The points X(70°S, 60°E) and Y(7°S, 60°E) lies on the surface of the earth.
(i) Illustrate the information in a diagram.
(ii) Find the distance between X and Y along the meridian.
[Take л = 22/7 and R = 6,400 km]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


        Calculations in the post by @greenestamps, giving the answer of 3520 km, are erroneous.


From the school, we know that the Earth circumference along the equator
is about 40,000 kilometers.

In this problem, we consider the Earth as spherical; so, its circumference
in any meridional section is the same 40,000 km.

Hence, under this assumption, the distance from the South pole to the North pole
along any meridian is half of 40,000 km, i.e. 20,000 km.

In this problem, the distance between the given points is 70° - 7° = 63°, measured as the difference of altitudes.

It is about one third of 180°, which represents the distance between the poles, expressed in degrees.

So, we can expect, that the distance between the given points is at least one third of 20,000 km.
i.e., at least 6666 km.


If you want to derive the value using the formula, you should write

        %28%2870-7%29%2F360%29%2A%282%2Api%2AR%29 = %28%2870-7%29%2F180%29%2A%2822%2F7%29%2A6400 = %2863%2F180%29%2A%2822%2F7%29%2A6400 = %287%2F20%29%2A%2822%2F7%29%2A6400 = %2822%2F20%29%2A6400 = 22*320 = 7040 kilometers.

It is the precise answer (under the given values and assumptions),
and it is consistent with my crude evaluation above.

---------------------------

Solved, with explanations.