Help with maxima question (1 Viewer)

excellence1 · Nov 24, 2015

A picture frame has a border of 2cm at the top and bottom and 3cm at the sides. If the total area of the border is to be 100cm ^2, find the maximum area of the frame.

Help plz

BLIT2014 · Nov 24, 2015

Firstly, we need to find two different areas.
The area of the picture and the area of the border.

Area of Picture = (x-6)(y-4)

Border Area = xy-(x-6)(y-4)
= 6y+4x-24
=100
y = (124-4x)/6
= (62-2x)/3

Area of Picture = (x-6)(y-4)
=(x-6)((62-2x)/3 -4)
= (x-6)(50-2x)/3
= (-2/3) ( x^2-31x+150)
= (-2/3) ( x^2-2 (31/2) x + (31/2)^2+150-(31/2)^2)
= (-2/3) ( (x-16.5)^2 +150-(31/2)^2 )
=(2/3)( -150+(31/2)^2 ) - (2/3)(x-16.5)^2

Therefore, the maximum of A, x=16.5,
Hence A = (2/3)( -150+(31/2)^2 )
= 60.17 cm^2

Misc workings:

3*2=6
2*2=4

InteGrand · Nov 24, 2015

excellence1 said:
A picture frame has a border of 2cm at the top and bottom and 3cm at the sides. If the total area of the border is to be 100cm ^2, find the maximum area of the frame.

Help plz

$Let the frame have dimensions $x$ (horizontal dimension) and $y$ (vertical dimension). (All lengths in fm and all areas in $\text{cm}^2$.) Then the area of the frame is $A = xy$. Now, drawing a diagram, we see that the area of the border is: $2\times \2x + 2\times 3\left(y-4 \right)$ (just adding up areas of rectangles). This this area must equal 100, we must have $2 \times 2x + 2\times 3\left(y-4\right) = 100$.$

$So the problem is now$

$$Maximise $A = xy$ subject to the constraint $2\times 2x + 2\times 3(y-4) = 100$ You should be able to do it from here (rearrange the constraint equation to get one variable in terms of the other and sub. this into the equation for $A$, then you just need to optimise a one-variable function, which you have probably learnt how to do.)$.$

$\textbf{Note.} My method here is for finding the max. area of the entire frame as implied by the question wording. BLIT2014's method finds the area of the picture in this maximal area frame. The two answers will just differ by 100 (the area of the border). \color{white}{So adding 100 to BLIT2014's method's answer will get you the frame's maximum area.}$

excellence1 · Nov 25, 2015

Thank you so much. I can know see where i went wrong

Help with maxima question (1 Viewer)

excellence1

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BLIT2014

The pessimistic optimist.

InteGrand

Well-Known Member

excellence1

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