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
