A clothing manufacturer uses the model a=f+4−−−−√−36−f−−−−−√ to estimate the amount of fabric to order from a mill. In the formula, a is the

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1. Questions:

   A clothing manufacturer uses the model a = √(f + 4) – √(36 – f) to estimate the amount of fabric to order from a mill. In the formula, a is the number of apparel items (in hundreds) and f is the number of units of fabric needed. If 400 apparel items will be manufactured , how many units of fabric should be ordered?

   
   

   Answer:

   32 units of fabrics

   Step-by-step explanation:

   Given

   [tex]a = \sqrt{f + 4} – \sqrt{36 – f}[/tex]

   Required

   Find f when a  = 4

   Substitute 4 for a

   [tex]4 = \sqrt{f + 4} – \sqrt{36 – f}[/tex]

   Rewrite as:

   [tex]\sqrt{36 – f} + 4 = \sqrt{f + 4}[/tex]

   Square both sides

   [tex](\sqrt{36 – f} + 4)^2 = (\sqrt{f + 4})^2[/tex]

   [tex](\sqrt{36 – f} + 4)^2 = f + 4[/tex]

   [tex]36 – f + 8\sqrt{36 – f} + 16 = f + 4[/tex]

   Collect Like Terms

   [tex]8\sqrt{36 – f}= f +f+ 4 – 36 -16[/tex]

   [tex]8\sqrt{36 – f}= 2f -48[/tex]

   Divide through by 2

   [tex]4\sqrt{36 – f}= f -24[/tex]

   Square both sides

   [tex]16(36-f) = (f – 24)^2[/tex]

   [tex]16(36-f) = f^2 – 48f + 576[/tex]

   [tex]576-16f = f^2 – 48f + 576[/tex]

   [tex]-16f = f^2 – 48f[/tex]

   Collect like terms

   [tex]f^2 – 48f + 16f = 0[/tex]

   [tex]f^2 -32f = 0[/tex]

   Factorize

   [tex]f(f – 32) = 0[/tex]

   [tex]f = 0[/tex] or [tex]f = 32[/tex]

   f can not be 0 because some units must be ordered.

   So, f = 32

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