# Calculus Problem - Optimization

• Nov 26th 2012, 04:08 PM
ktiama
Calculus Problem - Optimization
Hi i was wondering if anyone could help me with this calculus optimization problem i know how to do most optimization problems but this one stumps me.

Some lines through the point (2,1) cut off a triangle in the 1st quadrant. Find the x - intercept of the line with the property that the sum of its x and y intercepts is a minimum.

*What i know so far is that the equation of any line is y = mx + b
*The x intercept would be when y = 0 so 0 = mx + b
*The y intercept would be when x = 0 so y = b essentially
*I am not sure if this would be true but the equation of any line on that point would be 1 = 2m + b

I am not sure what to do from this point.

Help would be greatly appreciated :D
• Nov 26th 2012, 04:42 PM
ktiama
Re: Calculus Problem - Optimization
if i find the y intercept it would be y = b
if i find the x intercept it would be x = -b/m
so the sum would be b - (b/m)
which still doesnt make sense do i solve for m in the other equation?
• Nov 26th 2012, 05:06 PM
skeeter
Re: Calculus Problem - Optimization
the line through $(2,1)$ has equation ...

$y - 1 = m(x - 2)$

the y-intercept is $y_0 = -2m+1$

the x-intercept is $x_0 = 2 - \frac{1}{m}$

the sum of the intercepts is ...

$S = -2m + 3 - \frac{1}{m}$

Determine the value of the slope $m$ (remember it must be negative) that minimizes the sum, $S$ ... then find the x-intercept.
• Nov 26th 2012, 06:49 PM
ktiama
Re: Calculus Problem - Optimization
thank you so much skeeter