tangent line: a car's headlights

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stars584

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A car is traveling at night along a highway shaped like a parabola with its vertex at the origin. The car starts at a point 100 at west and 100 at north of the origin and travels in an easterlydirection. There is a statue located 100 at east and 50 m north of the origin. At what point on the highway will the car’s headlights illuminate the statue?

Car @ (-100,100) and Statue @ (100, 50).

Headlights is tangent which is y=2x

The the equation of the curve you see is just y=a x^2

(-100,100)
100=(-100)^2a
100=10,000a
a=0.01


Is this correct?

Can you guide me to the right direction?
 
You have your relationship between x and y, y = a*x^2

You almost have a slope, f'(x) = 2*a*x. You missed the 'a'.

You have a point, (100,50).

What's the point-slope form of a line? y-50 = 2*a*x*(x-100)

Substitution is available: a*x^2 - 50 = 2*a*x*(x-100)

Your remaining tasks are to solve the quadratic equation and to decide if both answers are valid.

Let's see what you get.
 
tangent

You have your relationship between x and y, y = a*x^2

You almost have a slope, f'(x) = 2*a*x. You missed the 'a'.

You have a point, (100,50).

What's the point-slope form of a line? y-50 = 2*a*x*(x-100)

Substitution is available: a*x^2 - 50 = 2*a*x*(x-100)



Ok, solving the above equation will give you a(x^2) -200ax +50

Now do I plug in the points??
 
For some mysterious reason, your equation disappeared. Put it back, then follow the instructions you have been given already. I'll repeat it for convenience.

Your remaining tasks are to solve the quadratic equation and to decide if both answers are valid.

Let's see what you get.
 
tangent line reply*******Please help**********

Ok lets see if I got it this time

CUrve line : y=ax^2
Tangent line : y=2ax

Point on curve: (100,-100)

a is determined to be a= 1/100 since 100=a(-100)^2

other point, (100,50).
a*x^2 - 50 = 2*a*x*(x-100)

THerefore,

(1/100)*x^2 - 50 = 2*(1/100)*x*(x-100)

(1/100)*x^2 = 1/50x^2-2x + 50

= (1/100)*x^2-2x +50

Is this correct???
 
Please help finishing this problem.

The equation of the headlights is y-100 = -2(x-(-100))

y = -2x-100

now to determine the point where the headlights beams at the statue I have the equation

The pt of the car is (x1,y1^2/100)

50 - ((y1)^2)/100 = x1/50 * (100-x1)


How can I solve this
 
Re: Please help finishing this problem.

stars584 said:
The pt of the car is (x1,y1^2/100)
Click to expand...
Whoops.

The pt of the car is (x1,y1)
The pt of the car is (x1,(x1^2)/100)

Substitute again and perhaps it will be easier to solve, having now only one variable.
 
50 - ((x1)^2)/100 = x1/50 * (100-x1)
50- x1^2/100 = 2x1 - x1^2/50
(100)*(50- x1^2/100 = 2x1 - x1^2/50)

5000- x1^2 = 200x1 -2x1

5000= 200x1 - x1^2

5000= x1(200-x1)

x1 = 5000 or -4800

is this correct
 
Dear stars,

Part of the study of mathematics is gaining confidence in yourself.

In this particular case, though you have been bold enough to show your work, you simply have not committed to your knowledge. Did you do the algebra right or not? Have you considered the Domain or not? Have you checked your answers or not? Did you make arithmetic errors or not? You can check all these things and gain some personal confidence in the result.

So, you tell me if it is correct, and tell me how you know that. First, you may wish to examine the Domain. How many cars are there? Can there be two solutions?
 
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