Question

1)Determine the centre and the radius of a circle whose equation is x^2+2x=16y-y^2

2)Expland (1+1/3x)^5 term in x^3.Hence use the expansion above to find the approximate value of (1.01)^5

3)Wekesa and Ndinda working together can cultivate a piece of land in 6 days.Wekesa alone can complete the work in 15 days.After the two had worked for 4 days,Wekesa withdrew the services.Find the time taken by Ndinda to complete the remaining work.

4)The position of two towns A and B on the earths surface are A(45°N,20°W) and B(45°N,150°E).Taking R=6370km,find the distance between A and B along the.parallel of latitude

5)The position of two towns on the surface of the earth are given as A(30°S,20°W) and B(30°S,80°E).Find;
a)The difference in longitude
b)The distance between the two towns along a parallel of latitude in:
i)km(take the radius of the earth as 6370km
ii)nm
c)Find the local time in town B when it is in 2:00pm in town A

Answer 1

[AI-generated answer — Bot GPT 3.5]

1)

To determine the center and radius of the circle, we first need to rewrite the equation in standard form:
x^2 + 2x = 16y - y^2
Rearranging, we get:
x^2 + 2x + y^2 - 16y = 0
Completing the square for both x and y terms:
(x+1)^2 - 1 + (y-8)^2 - 64 = 0
(x+1)^2 + (y-8)^2 = 65
Comparing with standard form of circle equation:
(x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius.
Therefore, the center of the circle is (-1, 8) and the radius is √65.

2)
Expanding (1+1/3x)^5 using the binomial theorem, we get:
1 + 5/3 x + 5*4/2*3 * (1/3)^2 x^2 + 5*4*3/3*2*1 * (1/3)^3 x^3 + ...
= 1 + 5/3 x + 10/9 x^2 + 10/27 x^3 + ...

Hence, to approximate (1.01)^5, we substitute x = 0.01 into the expansion above:
≈ 1 + 5/3*0.01 + 10/9*(0.01)^2 + 10/27*(0.01)^3
≈ 1 + 0.0167 + 0.0011 + 0.00037
≈ 1.01717

3)
Let the work rate of Wekesa be W and Ndinda be N.
Given that W + N = 1/6 (work done in a day) and W = 1/15 (Wekesa's work rate).
After 4 days, 4(W + N) + 6N = 1 (total work) => 4/6 +6N = 1.
Solving for N, we get N = 1/6.
Therefore, Ndinda can finish the remaining work in 6 days.

4)
To find the distance between A and B along the parallel of latitude, we use the formula for the distance between two points on a sphere:
Distance = R * θ, where θ is the central angle between the two points in radians.
Given that the latitude of A and B are the same, the central angle θ = 130° = 130*π/180.
Distance = 6370 * 130π/180 = 45433.3 km.

5)
a) The difference in longitude = 80°E - 20°W = 100°.
b)
i) Distance along the parallel of latitude = R * θ = 6370 * (100π/180) ≈ 35027.7 km.
ii) Distance in nautical miles = 35027.7 / 1.852 ≈ 18920.9 nm.
c) To find the local time in town B, we need to take into account the 60° per hour time difference for every longitudinal degree:
Time difference = 100 * 60 = 600 minutes = 10 hours.
Therefore, the local time in town B when it is 2:00 pm in town A is 12:00 pm.