minimize the area of the billboard with interior space of...

[lalala123]

hi
i am working on a textbook problem and I cant figure out, and I have a similar assignment problem

A billboard is to be made with an interior space for a poster of area 100m^2 and margin of 2m at the top and bottom and a margin of 4m at the sides. Find the outside dimensions of the billboard to minimize its total area.

so assuming y is the height and x is the length.

since area=length*height
A=l*h and area=100 x*y=100 so y=100/x so
A=(8+x)*(4+100/x)
=32+800/x+4x+100
=4x+800/x+132
4(x+200/x+33)

so now do I take the derivative of this:
A'(x)=4(1+200/x^2)
set it =0 so x=sq root 200

but this is not the right answer, can someone help me what i did wrong?

the final answer is sposed to be length=8+10 sq root2
and height= 4=5 sq root 2.
Can someone help please!!

 

[pka]

YOU ARE CORRECT!\(\displaystyle \L
x = \sqrt {200} = 10\sqrt 2 \quad \Rightarrow \quad y = \frac{{100}}{{10\sqrt 2 }} = 5\sqrt 2\)

 

[tkhunny]

Re: minimize the area of the billboard with interior space o

so assuming y is the height and x is the length.

My alarms went off the moment I read this.

y is the height of what? Interior or exterior?
x is the width of what? Interior or exterior?

Write clear, concise, and COMPLETE definitions. This practice will save you.

It appears you have defined x and y as deminsions of the internal area. You then set up the problem considering the dimensions of the external area. Finally, you solved for some vaue of x, which is the internal dimension again. It was only by magic that you managed the correct answer.

Really, make up your mind...

Try:

y is the height of the internal area
x is the width of the internal area

y+4 is the height of the external area
x+8 is the width of the external area

Now set it up and solve it.

Don't forget to go back and read the problem statement. Make sure you are answering the right question.

 

[daon]

Re: minimize the area

hi
i am working on a textbook problem and I cant figure out, and I have a similar assignment problem

A billboard is to be made with an interior space for a poster of area 100m^2 and margin of 2m at the top and bottom and a margin of 4m at the sides. Find the outside dimensions of the billboard to minimize its total area.

so assuming y is the height and x is the length.

Area= length*height
=(4+y)*(8+x)
=32+4x+8y+xy


and now im confused.. the final answer is sposed to be length=8+10 sq root2
and height= 4=5 sq root 2.
Can someone help please!!

Well, Area is A = xy = 100
So, x = 100/y

Then the outside area is A = (x+8)(y+4)
A = (100/y + 8)(y+4)
= 132 + 400/y + 8y
dA/dy = -400/y^2 + 8 = 0
so, y = \(\displaystyle \sqrt{50} = 5\sqrt{2}\).

Thus, x = \(\displaystyle \frac{100}{\sqrt{50}} = 10\sqrt{2}\)
Add your original margins and thats it.