The lion first runs 200 yards at 20 deg. south of west, then 300 yards at 40 deg. west of north, and finally 400 yards at 10 deg east of north at which point it catches up to the deer. How far from the starting point, and in what direction is the point where the lion and the deer meet?

D = 200yds[200o] + 300yds[130o] + 400yds

[80o].

X = 200*Cos200 + 300*cos130o + 400*Cos80
= -311.3 Yds.

Y = 200*sin200 + 300*sin130 + 400*sin80 = 555.3 Yds.

Tan Ar = Y/X = 555.3/-311.3 = -1.78391
Ar = -60.73o = Reference angle.
A = -60.73 + 180 = 119o CCW = 29o W. of N. = Direction.

D = Y/sin A = 555.3/sin119 = 635 Yds. @
29o W. of N.

To find the point where the lion and the deer meet, we can break down the lion's movements into its north and west components.

First, let's calculate the north and west components of the lion's movements:

For the first movement, the lion runs 200 yards at 20 degrees south of west. To find the west component, we multiply the distance (200 yards) by the cosine of the angle (20 degrees south of west). The west component is -200 * cos(20) = -186.93 yards.

To find the north component, we multiply the distance (200 yards) by the sine of the angle (20 degrees south of west). The north component is 200 * sin(20) = 68.50 yards.

For the second movement, the lion runs 300 yards at 40 degrees west of north. To find the west component, we multiply the distance (300 yards) by the sine of the angle (40 degrees west of north). The west component is -300 * sin(40) = -192.97 yards.

To find the north component, we multiply the distance (300 yards) by the cosine of the angle (40 degrees west of north). The north component is 300 * cos(40) = 229.38 yards.

For the third movement, the lion runs 400 yards at 10 degrees east of north. To find the north component, we multiply the distance (400 yards) by the cosine of the angle (10 degrees east of north). The north component is 400 * cos(10) = 396.93 yards.

To find the west component, we multiply the distance (400 yards) by the sine of the angle (10 degrees east of north). The west component is 400 * sin(10) = 69.09 yards.

Now, let's sum up the north and west components:

North component = 68.50 + 229.38 + 396.93 = 694.81 yards.
West component = -186.93 - 192.97 + 69.09 = -310.81 yards.

To find the distance from the starting point to the point where the lion and the deer meet, we use the Pythagorean theorem:

Distance = sqrt((North component)^2 + (West component)^2)
= sqrt((694.81)^2 + (-310.81)^2)
= sqrt(482159.16 + 96656.36)
= sqrt(578815.52)
= 760.45 yards (rounded to two decimal places)

To find the direction of the point where the lion and the deer meet, we can use trigonometry. The angle can be determined using the tangent function:

Angle = arctan(West component / North component)
= arctan(-310.81 / 694.81)
= arctan(-0.4466)
= -23.11 degrees (rounded to two decimal places)

Therefore, the point where the lion and the deer meet is approximately 760.45 yards away from the starting point, and it is approximately 23.11 degrees south of west from north.