The closest distance from a point to a line is its perpendicular distance to the line
The time it would take to be closest to Mt Rainier, is approximately 5.12 minutes
The reason why the value is correct is given as follows:
The given parameters are;
The direction of flight = 125km south and 135 km east
The location of Mt. Retainer = 56 km east and 40 km south of JBLM
The speed of flight = 800 km/hr
The slope of the travel path, m, is given as follows;
- [tex]m = -\dfrac{125}{135} = -\dfrac{25}{27}[/tex]
The equation of the path is therefore;
[tex]y + 125 = -\dfrac{25}{27} \times (x - 135)[/tex]
- [tex]y = -\dfrac{25}{27} \times (x - 135) - 125[/tex]
The required slope for the perpendicular distance from the travel path to the Mt Rainier, m', is therefore;
- [tex]m' = -\dfrac{1}{m}[/tex]
Which gives;
- [tex]m' = -\dfrac{1}{-\dfrac{25}{27}} = \dfrac{27}{25}[/tex]
The equation of the line is therefore;
[tex]y + 40 = \dfrac{27}{25} \times (x - 56)[/tex]
- [tex]y = \dfrac{27}{25} \times (x - 56) - 40[/tex]
The coordinates of the point where the two lines meet is therefore;
- [tex]\dfrac{27}{25} \times (x - 56) - 40 = -\dfrac{25}{27} \times (x - 135) - 125[/tex]
Solving with a graphing calculator gives;
x ≈ 50.09
y ≈ -46.38
- The magnitude of the distance in km is, d = [tex]\sqrt{(-46.38^2 + 50.09^2)} \approx 68.267[/tex]
The time it will take to be closest to Mt. Rainier, t, is given as follows;
- [tex]t \approx \dfrac{68.267 \, km}{800 \, km/hour} \approx 8.53 \times 10^{-2} \, hour[/tex]
The time it would take to get closest to Mt Rainier, t ≈ 8.53 × 10⁻² hours
[tex]The \ time \ in \ minutes, \, t = 8.53 \times 10^{-2} \ hour\times \dfrac{60 \, min}{hour} \approx \underline {5.12 \ minutes}[/tex]
Time it would take to be closest to Mt Rainier in minutes, t ≈ 5.12 minutes
Learn more about distance from a line to a point here:
https://brainly.com/question/1597347
