Please see below.

A pilot is capable of flying a plane with mass 6500kg, in a horizontal circle of radius r, at a constant speed of 75m/s. This is achieved by flying the jet with its wings at 35° to the horizontal. Calculate the magnitude of the lift force acting on the plane and hence calculate the radius r. 

Define our known variables:                     

m =  6500kg, v = 75ms, g = 9.81ms, r = ?

a) We know that there are only two forces acting on the plane, its weight (W) vertically downwards and the lift force (L) perpendicular to the wings of the plane. It always helps to draw a picture!

Since the plane is not accelerating vertically, Newton’s first law tells us that the forces in the vertical plane must sum to 0.

=> W – L_v = 0

=> W = L_v      where W = mg = 6500 x 9.81 = 63765N

=> L_v = 63765N

From this we can then work out the magnitude of the lift since the force acts at 35° to the horizontal using trigonometry. In this case we have an angle, an adjacent side and we are looking for the hypotenuse so we can use the CAH rule.

H = A ÷ C

=> L = L_v ÷ cos(θ)

=   63765 ÷ cos(35)

=> L = 77842.69165N ≈ 78000N

b) We know we are working with circular motion so there must be acceleration towards the centre of the circle and hence there is a resultant force in the same direction. In this example, the centripetal force is caused by the lift from the wings of the plane.

When trying to work out the radius we know we are dealing with a horizontal circle which means the weight of the plane has no component in the horizontal direction so the resultant force is purely from the lift so we can resolve the force horizontally using the SOH rule.

O = H x sin(θ)

=> L_h = L x sin(θ)

=  78000 x sin(35)

=> L_h = 44738.96204

Looking at the formula sheet, we can see we have enough variables to calculate the radius using:

F = (mv²) / r

Which we can rearrange to make r the subject and solve.

r = (mv²) / F

= (6500 x 75²) / 44738.96204

=> r = 817.2406854m ≈ 820m