Almost got it, but not quite.

**weekendclimber** · March 3rd 2010, 10:21 AM

So, I'm teaching myself calculus using the Open Course Ware at MIT and a few other online resources. I'm new to this forum and am hoping that if I get stuck anywhere I can bounce questions off someone here.

I think have most of this problem figure out, but I'm running into a dead-end. This shows up in the Problem Set 1, for course 18.01 'Single Variable Calculus'. Anyway here's the problem:

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 − x^2 (1000 minus x squared). An ant climbs up the mound starting from ground level (y = 0). At what height y does the ant begin to see the tower?

Now, I can derive the equation of the tangent line that meets at point (0, 1100) from the equation y = 1000 - x^2 as y = 1100 - 2x. Since there is only one point where these two equations meet I thought I should be able to use the quadradic equation to solve 0 = x^2 - 2x + 100 but I get an imaginary number ( +- sqrt( -396 )). Can anyone tell me where I'm off track? Thanks

**Archie Meade** · March 3rd 2010, 10:48 AM

Originally Posted by weekendclimber

So, I'm teaching myself calculus using the Open Course Ware at MIT and a few other online resources. I'm new to this forum and am hoping that if I get stuck anywhere I can bounce questions off someone here.

I think have most of this problem figure out, but I'm running into a dead-end. This shows up in the Problem Set 1, for course 18.01 'Single Variable Calculus'. Anyway here's the problem:

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 − x^2 (1000 minus x squared). An ant climbs up the mound starting from ground level (y = 0). At what height y does the ant begin to see the tower?

Now, I can derive the equation of the tangent line that meets at point (0, 1100) from the equation y = 1000 - x^2 as y = 1100 - 2x. Since there is only one point where these two equations meet I thought I should be able to use the quadradic equation to solve 0 = x^2 - 2x + 100 but I get an imaginary number ( +- sqrt( -396 )). Can anyone tell me where I'm off track? Thanks

Your mistake is to use the slope of the line as 2.

the line equation you are using does not intersect the parabola,
hence the reason for your complex answers.

The derivative (instantaneous slope) of the parabola is -2x.

The slope of the line you are looking for is "m".

$1100-mx=1000-x^2$

$x^2-mx+100=0$

$x=\frac{m\pm\sqrt{m^2-400}}{2}$

Therefore, to obtain a single positive solution for x

$m^2=400$

$m=\pm20$

The ant can can climb up either side, hence the positive and negative m.

**Opalg** · March 3rd 2010, 11:11 AM

Originally Posted by weekendclimber

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 − x^2 (1000 minus x squared). An ant climbs up the mound starting from ground level (y = 0). At what height y does the ant begin to see the tower?

At the point $(x_0,1000-x_0^2)$ on the parabola, the slope is $-2x_0$ and the equation of the tangent is $y - (1000-x_0^2) = -2x_0(x-x_0)$ . The condition for this line to pass through the point (0,1100) is $100-x_0^2 = -2x_0^2$ , or $x_0^2 = 100$ . So $x_0 = \pm10$ , and the corresponding value of y is $1000-100 = 900$ .

Originally Posted by weekendclimber

Now, I can derive the equation of the tangent line that meets at point (0, 1100) from the equation y = 1000 - x^2 as y = 1100 - 2x.

The mistake there is that you are using the same notation (x,y) to denote a fixed point on the parabola (the point where the tangent touches it), and a variable point on the tangent line. In my solution, I used $(x_0,y_0)$ for the point on the parabola, and (x,y) for a point on the tangent line. That way, you avoid confusing them.

**weekendclimber** · March 3rd 2010, 11:26 AM

That makes sense. Thank you for your help.

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Almost got it, but not quite.