a car is traveling along a highway shaped like a parabola with its vertex at the origin. the car starts at 100 miles north and 100 miles west. there is a statue at 100 miles east and 50 miles north. at what point will the cars headlights hit the statue?
y = a x^2
100 = a(10,000)
a = .01
so our parabola is y = .01 x^2
now a tangent to that parabola must pass through (100,50)
slope = .02 xp at parabola
y = (.02xp)x + b
50 = 2 xp + b
b = 50-2xp
but
yp = .01 xp^2
and
yp =.02xp^2 +50-2xp
.01 xp^2 = .02xp^2 + 50 - 2 xp
.01 xp^2 -2xp +50 = 0
xp^2-200xp+5000 = 0
xp = 100 +/- 50sqrt2
so about 70
if xp = 70
yp = .01x^2 = 49
let the equation of the parabola be
y = ax^2 , with (-100,100) on it
100 = a(-100)^2
100 = 10,000a
a = 100/10000 = 1/100
so y = (1/100)x^2
The car's headlights will hit the point (100,50)
when the light-line is tangent to the parabola at the point (a,b)
dy/dx = (1/50)x
at (a,b)
dy/dx = a/50
also b = a^2/100
slope of tangent:
(b-50)/(a-100)
so (b-50)/(a-100) = a/50
b - 50 = (a/50)(a-100)
a^2/100 = a^2/50 - 2a + 50
times 100
a^2 = 2a^2 - 200a + 5000
0 = a^2 - 200a + 5000
a = (200 ± √20000)/2
= (200 ± 100√2)/2
= 100± 50√2
= appr 29.29 miles or 170.71 miles
if a = 29.29 , b = 8.57
if a = 170.71, b = 291.42
As the car approaches from the left, the SUPER-LIGHT will hit the statue when the car is
29.3 miles east and 8.6 miles north
This is silly question. The car's headlights would have to shine over 80 miles !!!
xp = 100 +/- 50sqrt2
so about 30 (NOT 70 AS I SAID)
if xp = 30
yp = .01x^2 = 9
To find the point where the car's headlights hit the statue, we can set up a coordinate system with the vertex of the parabola (origin) as the center. Let's assume the car's position is given by (x, y) coordinates.
1. The car starts at 100 miles north and 100 miles west, so its initial position is (-100, -100).
2. We need to find the equation of the parabola. Since we know the vertex is at the origin, the equation will have the form y = ax^2. To find the value of 'a,' we can use the fact that the car starts at 100 miles north and 100 miles west.
Plugging in the values for the car's initial position:
-100 = a * (-100)^2
-100 = 10000a
a = -100 / 10000
a = -0.01
Therefore, the equation of the parabola is y = -0.01x^2.
3. We know the statue's position is 100 miles east and 50 miles north, so its coordinates are (100, 50).
4. The car's headlights hit the statue when their y-coordinate becomes the same. So, we need to find the x-coordinate where y = -0.01x^2 is equal to 50.
-0.01x^2 = 50
Divide both sides by -0.01:
x^2 = -50 / -0.01
x^2 = 5000
x = sqrt(5000)
x ≈ 70.71 (rounded to two decimal places)
Therefore, the x-coordinate where the car's headlights hit the statue is approximately 70.71.
5. To find the y-coordinate at this point, substitute x into the equation y = -0.01x^2:
y = -0.01 * (70.71)^2
y ≈ -50
Therefore, the y-coordinate where the car's headlights hit the statue is approximately -50.
Hence, the car's headlights will hit the statue at the point (70.71, -50).