A rectangle is to have an area of 16 square inches. How do you find its dimensions so that the distance from one corner to the midpoint of a nonadjacent side is a minimum?
1 Answer
Explanation:
We can write the following equations:
lw=16lw=16
Draw a diagram of the line cutting through the rectangle and use the Pythagorean Theorem to say that the length of the segment can be found through:
f(l,w)=sqrt(l^2+(w/2)^2)f(l,w)=√l2+(w2)2
Using the area equation, we can make
l=16/wl=16w
Thus,
Simplify:
f(w)=sqrt(256/w^2+w^2/4)=sqrt((1024+w^4)/(4w^2))=(sqrt(w^4+1024))/(2w)f(w)=√256w2+w24=√1024+w44w2=√w4+10242w
It should be noted that the domain of this function, or the values for which
To find the minimum value, find the derivative of
f'(w)=((4w^3(2w))/(2(sqrt(w^4+1024)))-2sqrt(w^4+1024))/(4w^2)
=((4w^4)/sqrt(w^4+1024)-(2(w^4+1024))/sqrt(w^4+1024))/(4w^2)=(2w^4-2048)/(4w^2sqrt(w^2+1024))
=(w^4-1024)/(2w^2sqrt(w^4+1024))
Set the derivative equal to
w^4-1048=0=>w=root(4)1024=>w=4sqrt2
The derivative does not exist when
To find the extrema, find the function values for the endpoints of the domain,
Since
lim_(wrarr0)f(w)=oo
f(4sqrt2)=4
lim_(wrarroo)f(w)=oo
Since
Note that since the Pythagorean formula I created used