Min/Max word problems help appreciated

[sarahjohnson]

Hi so I don't understand how to do these types of word problems still. Can anyone walk me through finding the equation? I think I can do the rest by myself. Much thanks!


Link if it's too small http://postimg.org/image/7xjq1vgvr/

Also for d) it asks (d) Somewhere between d = 7 m and d = 14 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Find this exact value of d.

Is that the inflection point?

 

[DrPhil]

Hi so I don't understand how to do these types of word problems still. Can anyone walk me through finding the equation? I think I can do the rest by myself. Much thanks!

View attachment 3132
Link if it's too small http://postimg.org/image/7xjq1vgvr/

Also for d) it asks (d) Somewhere between d = 7 m and d = 14 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Find this exact value of d.

Is that the inflection point?

For the first part of the question, \(\displaystyle d\) is a parameter and the variable is \(\displaystyle x\). Write expressions for the square of the distance of point \(\displaystyle P = (x,\ d)\) from each light source. The intensity from each source is 1/(distance)^2, and you can just add the two intensities. Do they want you to combine the two terms over a common denominator? Probably don't have to for part (a), but it may be useful later.

Treat \(\displaystyle d\) as a constant and minimize with respect to \(\displaystyle x\) in the domain \(\displaystyle 0 \le x \le 14\)m.

EDIT: Technically, when you want the absolute minimum over a limited domain, you have to check the points where the derivative is 0 and also the boundaries of the domain.