1) a flag post 10m long is fixed on top of a towerefrom a point on horizon ground the angles of elevation of the top an bottom of thge flag post are 400m and 300n respectively calculate the height of the tower and the shortest vdistance from the point on the ground to the top of the flag post
2)a man walks directly from point a towards the foot of a tall bulding 240m away after covering 180m he observes that the angle of the top of the bulding is 450 determine the angle of elevation of the top of the bulding from a
3)find the position vector of point r which divides line mn internally in the ratio 2:3 take the position vectors of m and n to be m=(3/4/6)and n=(-5/-1/2)
1) Let the height of the tower be h. From the point on horizon ground, the distance to the bottom of the flag post is x (which is also the shortest distance to the top of the flag post).
Using trigonometry, we have:
tan(40) = (h/x+10) and tan(30) = h/x
Solving for h and x, we get:
h = 10[tan(40) - tan(30)] ≈ 6.00 m
x = h/tan(30) ≈ 10.39 m
Therefore, the height of the tower is approximately 6.00 m and the shortest distance from the point on the ground to the top of the flag post is approximately 10.39 m.
2) Let the angle of elevation of the top of the building from point A be α. Then we have:
tan(α) = (height of building) / (distance from A to building)
Using trigonometry, we have:
tan(45) = (height of building) / (240 - 180)
Solving for the height of the building, we get:
height of building = 60
Therefore, we have:
tan(α) = 60/240 = 1/4
Taking the inverse tangent, we get:
α ≈ 14.04°
Therefore, the angle of elevation of the top of the building from point A is approximately 14.04°.
3) Let the position vector of point R be r. Then we have:
r = (3/5)(n-m) + m
Note that the ratio 2:3 divides the line segment MN internally, which means that the vector from M to R is 2/5 of the vector from M to N. Therefore:
r - m = (2/5)(n-m)
Multiplying both sides by 5 and adding m, we get:
r = (3/5)(n-m) + m
Substituting the given values for m and n:
r = (3/5)(<-5-3>/<-1-4>/<2-6>) + <3/4/6>
Simplifying, we get:
r = <-9/5, -9/5, 12/5> + <3/4/6>
Combining like terms, we get:
r = <-3/20, -9/20, 102/20>
Therefore, the position vector of point R is approximately <-0.15, -0.45, 5.10>.