Parabolas and lines.

A hot air balloon takes off from the edge of a plateau. The balloon takes off in a path ( y = (-4/2500) x^2 + .8x). The land drops away from the take off point at a rate of 1 vertical foot per every 5 horizontal feet ( so it looks like -1/5 x). In the problem, it asks to find where the balloon lands, the angle between the positive x axis and the land (the land line starts at the origin), and what is the maximum height of the the balloon above the ground.

y = - (4 / 2500)x² + 0.8x)
y = - 0.0016x² + 0.8x

Time to Max. ht.: y' = 0:

y' = - 0.0032x + 0.8

- 0.0032x + 0.8 = 0
- 0.0032x = - 0.8
x = - 0.8 / - 0.0032
x = 250

y = - 0.0016(250)² + 0.8(250)
y = - 0.0016(62500) + 200
y = - 100 + 200
y = 100

Max.Ht. = 100 ft.
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Point of Landing: y = 0:

- 0.0016x² + 0.8x = 0
x(- 0.0016x + 0.8) = 0

If the product of two factors equals zero, then one or both factors equal zero.

x = 0

If - 0.0016x + 0.8 = 0
- 0.0016x = - 0.8
x = - 0.8 / - 0.0016
x = 500

The balloon will land 500 ft. from its takeoff point.
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Angle Drop = Θ

Tan Θ = 1/5
Tan Θ= 0.2
Θ = Tanˉ¹ (0.2)
Θ = 11.3°

Inclination Angle = Φ

Φ = 90 - Θ
Φ = 90 - 11.3
Φ = 78.7°
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