The size of the corner you need to cut in order for your box to have this area is 0.5 inches
<h3>Part 1: A. Express the new length of the long side of the note card, once the two corners are removed.</h3>
The base length is given as:
Length = 7
When the edges are removed, the new length becomes
New length = 7 - x - x
Evaluate
New length = 7 - 2x
<h3>Part 1: B. Express the new width of the short side of the note card, once the two corners are removed.</h3>
The base width is given as:
Width = 4
When the edges are removed, the new length becomes
New width = 4 - x - x
Evaluate
New width = 4 - 2x
<h3>Part 2: Write a function A(x) that defines the area of the bottom of the box, once the corners are removed and the sides are folded up.</h3>
The area of the bottom of the box is calculated as:
Area = New length * New width
This gives
Area = (7 - 2x) * (4 - 2x)
Rewrite as:
A(x) = (7 - 2x) * (4 - 2x)
<h3>Part 3: Set up an equation in standard form that will help you find the size (x)</h3>
The area is given as:
Area = 16
So, we have:
(7 - 2x) * (4 - 2x) = 16
Expand
28 - 14x - 8x + 4x^2 = 16
Rewrite as
4x^2 - 14x - 8x + 28 - 16 = 0
Evaluate the like terms
4x^2 - 22x + 12= 0
<h3>Part 3: Solve this equation and take note of any extraneous solutions</h3>
We have:
4x^2 - 22x + 12= 0
Expand the equation
4x^2 - 24x - 2x + 12 = 0
Factorize the equation
4x(x - 6) - 2(x - 6) = 0
Factor out x - 6
(4x - 2)(x - 6) = 0
Solve for x
x = 0.5 or x = 6
The value of x = 6 is too big for the dimensions of the box.
So, x = 6 is an extraneous solution for the equation
Hence, the size of the corner you need to cut in order for your box to have this area is 0.5 inches
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