# solving optimization word problem?

• Feb 1st 2011, 03:56 PM
aldricj
solving optimization word problem?

A rectangular banner has a red border and a white center. The banner has an 8 inch border at the top and the bottom, and it has a 6 inch border on each side. The total area of the banner is 27 ft squared or 3888 in squared. What should be the dimensions of the banner (x and y) if the area of the white center is to be a maximum?
• Feb 1st 2011, 04:13 PM
skeeter
Quote:

Originally Posted by aldricj

A rectangular banner has a red border and a white center. The banner has an 8 inch border at the top and the bottom, and it has a 6 inch border on each side. The total area of the banner is 27 ft squared or 3888 in squared. What should be the dimensions of the banner (x and y) if the area of the white center is to be a maximum?

let x = top length of white center
y = side length of white center

$(x+12)(y+16) = 3888$

$\displaystyle x+12 = \frac{3888}{y+16}$

$\displaystyle x = \frac{3888}{y+16} - 12$

maximize the area of the white center ...

$\displaystyle A = xy = \left(\frac{3888}{y+16} - 12\right)y$

take it from here?
• Feb 1st 2011, 04:50 PM
aldricj
okay, im pretty sure the next step is to take the derivative, but im not positive on how to do that. thanks for the response!
• Feb 1st 2011, 05:05 PM
skeeter
Quote:

Originally Posted by aldricj
okay, im pretty sure the next step is to take the derivative, but im not positive on how to do that. thanks for the response!

$A = \dfrac{3888y}{y+16} - 12y$

quotient rule for the first (rational) term; derivative of the second term is -12, correct?

set $\dfrac{dA}{dy} = 0$ and determine the value of y that maximizes $A$
• Feb 2nd 2011, 05:26 AM
aldricj
okay, i guess i should just say i need to see the step by step process to do this problem. im getting frustrated not being able to find the answer(Headbang)
• Feb 3rd 2011, 05:51 AM
skeeter
$A = \dfrac{3888y}{y+16} - 12y$

$\dfrac{dA}{dy} = 3888 \cdot \dfrac{(y+16)(1)-(y)(1)}{(y+16)^2} - 12 = 0$

$\dfrac{16}{(y+16)^2} = \dfrac{12}{3888}
$

$\dfrac{16}{(y+16)^2} = \dfrac{1}{324}$

$(y+16)^2 = 16 \cdot 324
$

can you finish?