Question

1. furniture manufacturing you are making a coffee table with a glass top surrounded by a cherry border. the glass is 3 feet by 3 feet. you want the cherry border to be a uniform width. you have 7 square feet of cherry to make the border. what should the width of the border be? 3 ft

Explanation:

Step1: Define the outer square's side

The glass is 3 ft by 3 ft, and the border has width \( x \). So the outer square (glass + border) has side length \( 3 + 2x \) (since \( x \) is added to both sides).

Step2: Calculate area of outer and inner squares

Area of inner square (glass) is \( 3 \times 3 = 9 \) sq ft.
Area of outer square is \( (3 + 2x)^2 \) sq ft.

Step3: Set up the equation for border area

The border area is outer area minus inner area, which equals 7 sq ft.
So, \( (3 + 2x)^2 - 9 = 7 \).

Step4: Expand and simplify the equation

Expand \( (3 + 2x)^2 \): \( 9 + 12x + 4x^2 - 9 = 7 \)
Simplify: \( 4x^2 + 12x - 7 = 0 \).

Step5: Solve the quadratic equation

Use quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4 \), \( b = 12 \), \( c = -7 \).
Discriminant: \( 12^2 - 4(4)(-7) = 144 + 112 = 256 \)
Square root of discriminant: \( \sqrt{256} = 16 \)
So, \( x = \frac{-12 \pm 16}{8} \).

Two solutions:
\( x = \frac{-12 + 16}{8} = \frac{4}{8} = 0.5 \)
\( x = \frac{-12 - 16}{8} = \frac{-28}{8} = -3.5 \) (discard, since width can't be negative).

Answer:

The width of the border should be \( 0.5 \) feet (or \( \frac{1}{2} \) foot).