A pilot of a downed airplane fires the emergency flare into the sky. The path of the flare is modelled by the equation h=-0.096(d-25)^2 +60 where h is the height of the flare in meters when its horizontal distance from where it was propelled is d meters. An emergency helicopter equipped with special binoculars has a line of sight to the spot where the flare was launched. The line of sight from the binocular is modelled by the equation 9x-10y=-14
Solve the system and give answers rounded to two decimal places. The line of sight from the binoculars spots the flare twice. How high was the flare closest to the ground when it was spotted the first time?
I have no idea how to do this as I only know how to solve with two different variables and not three. Do I sub x for d in the first equation? any help would be appreciated
what three variables? All you have to do is see where the line intersects the parabola. Calling the first one h and d instead of y and x doesn't introduce any new variables. They are just names to describe the curve.
9x-10y=-14
y = (9x+14)/10
So, to see where they intersect, solve
-0.096(d-25)^2 +60 = (9x+14)/10
see
http://www.wolframalpha.com/input/?i=-0.096(x-25)%5E2+%2B60+%3D+0.1(9x%2B14)
To solve the system of equations, you can substitute the value of one variable from one equation into the other equation. In this case, you can substitute the value of 'd' in terms of 'x' from the second equation into the first equation.
The second equation represents the line of sight from the binoculars, given by the equation 9x - 10y = -14. Rearranging this equation to solve for 'y' in terms of 'x', we get:
10y = 9x + 14
y = (9/10)x + 14/10
y = 0.9x + 1.4
Now, substitute this value of 'y' into the equation for the flare's height:
h = -0.096(d - 25)^2 + 60
Since the flare is spotted twice, you are looking for the values of 'd' and 'h' for the first occurrence. So, substitute 'd' in terms of 'x' into the equation:
h = -0.096((0.9x + 1.4) - 25)^2 + 60
Simplifying, we have:
h = -0.096(-23.1x - 23.6)^2 + 60
Now, use a graphing calculator or software to plot this equation and find the x-coordinate(s) where the flare's height is maximum (the first occurrence). These x-coordinates correspond to the horizontal distance 'd'. Determine the value of 'h' (flare's height) at this point to find the answer.
Alternatively, you can use calculus to find the maximum value of 'h' by taking the derivative of the equation:
dh/dx = -0.096(2)(-23.1x - 23.6) = 0
-0.192(-23.1x - 23.6) = 0
4.4406x + 4.4832 = 0
4.4406x = -4.4832
x ≈ -1.01
Substitute this value of 'x' back into the equation for 'h' to find the flare's height at this point:
h ≈ -0.096((-1.01)(0.9) + 1.4 - 25)^2 + 60
h ≈ -0.096(-24.04)^2 + 60
h ≈ -0.096(577.9216) + 60
h ≈ -55.651 + 60
h ≈ 4.349
Therefore, the flare's height closest to the ground when it was spotted the first time is approximately 4.35 meters.