What height does the ant start seeing the tower?

[pamparana]

Hello everyone,

This is not exactly my homework question. I was looking at the assignments on MIT open courseware page for Single variable calculus and came across this one:

Problem statement
Quirk is a flat planet. On the planet Quirk, a cell phone tower is a 100-foot pole on top of a green mound 1000 feet tall whose outline is described by the parabolic equation y = 1000 − x2. An ant climbs up the mound starting from ground level (y = 0). At what height y does the ant begin to see the tower?

Homework Equations

I guess I will need the derivative at some point: f'(x) = -2x. So, it has a negative slope.

The Attempt at a Solution

I am having trouble visualizing the problem. So, the curve meets the y axes at height 1000 and the pole is another 100 feet, so I have a line from (0, 1100) which will meet the curve at some point P. I have to find this point P. Is that correct?

I guess I will need to find the slope of this line but I am having trouble seeing how this could connect to the slope of the tangent line to the parabola at point P.

Thanks for any help you can give me.

/Luca

 

[Dick]

The coordinates of the point P will be (x,1000-x^2) for some x, right? Call Q the point at the top of the cell tower (0,1100). You want the line through PQ to be tangent to the parabola. The slope of the tangent is -2x. Set that equal to the slope you get from m=delta(y)/delta(x) using the points P and Q and solve for x.

 

[pamparana]

You want the line through PQ to be tangent to the parabola.

Thanks for your reply Dick. This is the bit that I had completely missed that this line would be tangent to the parabola.

Many thanks,

Luca