# Thread: Calculus Problem - Optimization

1. ## 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 2. ## 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?

3. ## Re: Calculus Problem - Optimization

the line through $\displaystyle (2,1)$ has equation ...

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

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

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

the sum of the intercepts is ...

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

Determine the value of the slope $\displaystyle m$ (remember it must be negative) that minimizes the sum, $\displaystyle S$ ... then find the x-intercept.

4. ## Re: Calculus Problem - Optimization

thank you so much skeeter

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