A wire 10 long is cut into two piece, one of length x and the other of length 10

Question

A wire 10 long is cut into two piece, one of length x and the other of length 10-x . Each piece is bent into the shape of a square. a. Find a function that models the total area enclosed by the two squares 2. b. What is the domain of your function? c. Find the value of x that minimizes the total area of the two squares 3. People have been making open-topped boxes for decades. Think about all those corer squares of paper being thrown out! Must they be wasted? Absolutely not! By taping them together, and putting the resultant structure on a desk, one can make a handsome pen-and-pencil holder, which will be a box with neither top nor bottom. (It will still hold pens and pencils, as long as it rests on the desk) 8.5 Write a function for the combined volume of an open-topped box plus a handsome pen-and pencil holder that can be made by cutting four squares from an 8.5 x 11 inch sheet of paper? a) b) What is the domain of your function? c) Use a graphing utility to find the maximum possible combined volume of an open-topped box and a handsome pen-and pencil holder?

Explanation / Answer

2) a) piece of 10 cm is cut into two pieces

one is x

and other is 10 - x

so, length of each side of square of 1st piece = x/4

length of each side of square of 2nd piece = ( 10 - x) / 4

area of 1st piece = x^2 / 16

area of 2nd piece = ( 10 - x)^2 / 16

a) total area A(x) = ( 2x^2 - 20x + 100 ) / 16

b) domain is all values of x > 0

( 0 , 10)

c) value of x that minimizes the area

x = -b / 2a

x = 20 / 4 = 5

value of x = 5 cm

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