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Last updated: February 19, 2017

 
Optimization Problem Using the Derivative To Solve For X
  Posted: 4 October 2014
Area of a Rectangle*
 

A=LW

Area = (Length)(Width)

*The alternative version A=bh is more commonly used: Area=(base)(height).

Imagine designing a book cover that will have margins on all sides around an area with text and graphics also called the print area of the cover. The book cover will have white space in the margins which are 1 inch wide and 1.5 inches high. The area with print inside of the margins will have a total area of 48 inches. What will be the length and width of the whole book cover to optimize the 48 inches of print area in the center of the book cover with the specified margins?

To complete this problem you will need to know the formula for the area of a rectangle which is area equals length times width.

Knowing that the margins are 1 inch wide and 1.5 inches high, you can write an equation for the area of the book cover using a quantity to represent the length and width of the book cover. Because 1.5+1.5=3 the length of the book cover will be represented by the quantity: (x+3); and because 1+1=2, the width of the book cover will be represented by the quantity: (y+2).

         
 
1
w
1
 
1.5
 
 
1.5
h
 
y
   
 
x
Print
x
 
y
1.5
 
 
1.5
 
1
1
 

 

Length (or h for "height") = (x+3)

& Width = (y+2);

therefore...

A= (x+3)(y+2) is the primary equation in solving this problem. The secondary equation is formed by knowing that the book cover has 48 inches of printing area. Therefore, the secondary equation is 48=xy.

Later in this problem the derivative of the equation for the area of the book cover will be needed. The Simple Power rule and the Constant rule are used to find the derivative of the area in steps below.

Domain of X Matters Because X is Solved For First and Y Depends on X

From looking at the secondary equation "xy=48", it is clear that x cannot be 0 or less than 0 because the length must be positive, since we don't count in negative inches, and the length multiplied by the width must equal the positive natural number of 48.

 

 

 
Step 1  
secondary equation of the area function
Write the secondary equation that will be a quantity in the area function we are going to make to solve the question of height and width of the book cover.
   
Step 2  
solving for y in the secondary equation
  • Solving the secondary equation for y results in the equation at left.
  • Input this y-value into the Area function for the y-value and then solve for x.
   
Step 3  
Writing an equation for the area of a book cover

 

Multiply the two quantities in parenthesis with each other by firsts, outers, inners, lasts (FoIL).
   
Step 4  
Area function
The 3rd term in the area function we are solving for has 144 in the numerator because 48 x 3 = 144. In step 5 simplify the area function.
   
Step 5  
step 5
Simplified area function. Next step is to combine like terms.
   
Step 6  
Area function
This is the midway point in solving the problem. The next step is to find the derivative of A.
   
Step 7  
derivative of the area function
To find the derivatie of A you would have to use the "simple power" and the "constant" rule. Using the derivative notation, the derivative of A looks like the equation at left.
   
Note about derivative notation: The Derivative notaton for A is dA/dx and it is used until you "take the derivative" and then the notation goes poof & disappears. Since the derivative is taken in step 7 the "dA/dx" notation goes poof & disappears and you set the derivative equal to zero and solve for x in Step 8.
   
Step 8  
Step 8 Set the derivative equal to zero and solve for x. Get the second term on the left side of the equals sign.
   
Step 9  
Set derivative equal to zero and solve for x Cross multiply.
   
Step 10  
Step 10

Divde both sides by 2.

   
Step 11  
step 11 Take the square root of both sides.
   
Step 12  
x-solution Though a square root has both a negative and a positive solution as shown as left separated by a comma, we already know that the domain of x can only be positive so we can disregard the negative value in the solution.
   
Step 13  
Step 13 Plug x into the value in the primary equation. The height of the book cover to the nearest tenth is 11.4 inches.
   
Step 14  
Step 14 To find the width of the book cover plug in the x-value solution, for the x that is substituted into the y-quantity of the primary equation to find the width.The width of the book cover to the nearest tenth of an inch is 7.6 inches.
   
Solution:

The book cover is 11.4 by 7.6 inches in order to have 48 square inches of printing area space with 1 inch vertical margins and 1.5 inch horizontal margins.