math problem?

Question

1/Picture link:http://img128.imageshack.us/my.php?image=heatsinktx7.jpg
We would like to maximize the area of the cooling fins in order to maximize cooling, but the board have an area of 40 cm2 .Find the dimensions of the board that would maximize the cooling provided by the fins.
2/picture:http://img116.imageshack.us/my.php?image=cellphonetowerssx2.jpg
Where should those towers be situated in order that the direct line-ofsight distance between them be least?
plz answer clearly.

Answer 1

1) The diagram shows 6 rectangles.
Let the horizontal sides be x and
vertical sides be y. The area of the board is 2x*3y.
The length of all the fins is 4x + 9y
We just have to maximize the length because the area of the board is 40 and the width of the fin is 2, which are constant.
Area = 2x*3y = 40
xy = 40/6
y = 20/3x --------------(1)
Length (L)= 4x + 9y
becomes L = 4x + 9(20/3x)
L = 4x + 60/x
dL/dx = 4 - 60/x^2
0 = 4 - 60/x^2
x = sqrt 15
x = 3.87
From (1)
y = 20/3(3.87)
y = 1.72


2) Let x = distance from tower 1
and y = distance from tower 2.
12/x = y/16 by similar triangle
xy = 192
y = 192/x ---------------(1)
y^2 = (192/x)^2 -------(2)

Let the distance between tower! and 2 be S.
S^2 = (x^2 + 12^2) + (y^2 + 16^2) ----------------(3)
Sub (2) into (3)
S^2 = x^2 + 12^2 + (192/x)^2 + 16^2
S^2 = x^2 + 12^2 + 36864x^-2 + 16^2

Differentiate :
2SdS/dx = 2x - 73728x^-3
dS/dx = 0, you don't have to worry about S
0 = x - 36864 x^-3
x^4 = 36864
x = 13.856
y = 192 / 13.856
y = 13.856 also.

Answer 2

Hello,

Let's look at number 1 first

Let x = distance to from tower 1 to Chalmer's mtn.
Let y = distance from Chalmer's Mtn to tower 2.

Now these two triangles are similar so x/12 = 16/y then y = 12*16 / x so y = 192/x Now in the left triangle the hypotenuse = sqrt(x^2 + 12^2) and in the right triangle the hypotenuse = sqrt((192/x)^2 + 16^2)

Adding these two values gives us the distance between the two towers. Now we must take the derivative .

d = sqrt(x^2 + 144) + sqrt( (192/x)^2 + 256))

I'll let you finish this.

Hope This Helps!!