Maths problem using 2 different elevations from a distance apart?

j = horizontal distance of kite from John

m = horizontal distance of kite from Mary

h = j∙tan(30°)

h = m∙tan(38°)

j∙tan(30°) = m∙tan(38°)

By the Pythagorean theorem, j=√(m²+90²), so

√(m²+90²)tan(30°) = m∙tan(38°)

(m²+90²)tan²(30°) = m²∙tan²(38°)

m²tan²(30°) +90²tan²(30°) = m²∙tan²(38°)

90²tan²(30°) = m²∙tan²(38°) - m²tan²(30°)

90²tan²(30°) = m²(tan²(38°) - tan²(30°))

m² = 90²tan²(30°)/(tan²(38°) - tan²(30°))

m = 9.7

h = 98.7tan(38°)

= 77.1

The kite is 77.1 meters above level ground.

John (0, 0, 0)

Mary (90, 0, 0)

kite (90, 98.7, 77.1)

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Thank you, Sir! May I have another?

Are you ready for this?

You have three triangles to solve.

The first is the triangle Mary, John, point directly below the kite.

The distance from Mary to John is 90m.

Call the distance from Mary to the point directly below the kite M and likewise the distance from John to the same point J. So, we have a triangle with sides 90, M, J. This is the triangle drawn flat on the ground. The other two triangles stand at right angles to it.

Call the height, h. The second triangle has adjacent sides M and h. The angle between the two sides being a right angle. The angle opposite h is 38 degrees. Hence,

h = M * tan(38) . . . . (1)

The third triangle has adjacent sides J and h. The angle between them being a right angle. The angle opposite h is 30 degrees. Hence,

h = J * tan(30) . . . . (2)

Therefore M * tan(38) = J * tan(30)

and

M / J = tan(30) / tan(30)

This gives you the ratio of the sides M and J of the first triangle (the one that drawn flat on the ground)

Let x be the angle between the sides of length J and 90. Then

x = inverse sine of (M / J).

Now, tan(x) = M / 90, hence

M = 90 * tan(x).

Substituting this into equation (1) above gives

h= 90 * tan(x) * tan(38)

Likewise

cos(x) = 90 / J, so

J = 90 / cos(x)

Substituting into equation (2) gives

h = 90 * tan(30) / cos(x)

You have two angles and a distance

Find the third angle

Then use law of sines to find the distance that John or Mary is from the kite.

Then, draw an altitude from the kite to the ground, and find that height with pythagorean theorem

Put Mary at the origin, John 90 units left of O. (Draw...) The 'base' of Mary's triangle with the kite is: x = 95/(tan30) - 90 = 74.54 Therefore, the height of the kite is: h = 74.54(tan38) = 58.24 m