A). The rectangle of maximum area that could be isncribed inside the triangle always has dimensions equal to half the lengths of the sides adjacent to the rectangle. Prove that this is true for any right triangle.
B) Prove that any cylindrical can of volume k cubic units that is to be made using a minimum amount of material must have the height equal to the diameter.
OK>>
I've been working on part 1 so far.. trying to, so i have:
I drew a triangle right angled, labelled points..
Triangle ABC has a rectangle in it labelled by EFD
B has the right angled and F is across from it. E is in the middle of the AB line and D is in the middle of the BC line.
Now for the points:
A(y,0)
E(0.5y,0)
D(0.5x,0)
C(x,0)
so y=mx+b
0=m and 0=b when i plugged them in
so my eqtn becomes y=x
Max Area of Rectangle
A(x)=xy
=x(x)
=x^2
A'(x)=2x
A'(x)=0 when x=0
A(0)=0
And from here, my proving did not go so well. ^.^ what should I do next?
this is just like that problem you did with the 5 and 12 sided legs of a right triangle.
let point A be on the y-axis with coordinates
point B is at the origin
point C is on the x-axis with coordinates
note that and are both positive constants
the equation of the line between A and C is
F is the point on the line AC
BE = y ... BD = x
area of the rectangle is
area of the rectangle ,
find and determine the value of x that maximizes R
you should get ... the x-value of the midpoint of AC
For part B)
What does the question mean by the height must equal the diameter for the volume of a cylindrical can using the min amount of material?
How can the height equal the diameter? I'm looking at one of my previous questions at the moment and I can see that the radius is 43 so the diamater is 86 and the height is 172.. 172 does not equal 86.
so what do they mean by this exactly?