A cyclist rode from town A to town B, 14km apart on a bearing of N 40 degree E. At B he rode due East to another town C, 19km away from B. Calculate, correct to the nearest whole number a) the distance between A and C.
b) how far North of A is B.
All angles are measured CW from +y-axis.
AC = AB + BC = 14km[40o] + 19km[90o].
X = 14*sin40 + 19*sin90 = 28 km.
Y = 14*Cos40 + 19*Cos90 = 10.7 km.
a. AC = sqrt(X^2+Y^2).
b. TanA = X/Y.
b. Cos40 = d/14.
d = 14*Cos40 = 10.7 km = Vertical distance from A.
To solve this problem, we will use the concept of trigonometry and vector addition.
a) To find the distance between town A and C, we can break down the motion of the cyclist into two components: North-South and East-West.
First, let's calculate the North-South component. Since the bearing is N 40° E, we can consider the North-South direction as the vertical component (opposite to the bearing). To find this component, we calculate the sine of the bearing angle:
North-South component = 14 km * sin(40°)
Next, let's calculate the East-West component. Since the bearing is N 40° E, we can consider the East-West direction as the horizontal component (adjacent to the bearing). To find this component, we calculate the cosine of the bearing angle:
East-West component = 14 km * cos(40°)
Now, let's find the total North-South displacement. The cyclist traveled 19 km East from town B to town C. Since both town B and town C are on the same latitude, the North-South displacement remains the same.
Total North-South displacement between town A and C = North-South component + North-South displacement from B to C
= North-South component + North-South component (from B to C)
= 2 * North-South component
Finally, we can find the distance between town A and C by using the Pythagorean theorem:
Distance between A and C = sqrt((Total North-South displacement)^2 + (East-West component)^2)
b) To determine how far North of A town B is, we need to find the North-South displacement from A to B.
North-South displacement between town A and B = North-South component
Now, let's compute the values:
North-South component = 14 km * sin(40°)
East-West component = 14 km * cos(40°)
Total North-South displacement between town A and C = 2 * North-South component
Distance between A and C = sqrt((Total North-South displacement)^2 + (East-West component)^2)
North-South displacement between town A and B = North-South component
By plugging in the values and performing the calculations, you will get the answers to the nearest whole number.