Question

If the cost see for manufacturing X units of a certain product is given by C=X squared +25X +40 find the number of units manufactured at a cost of 9390

Answer 1

Answer:

  85 units

Step-by-step explanation:

Fill in the given value and solve for x.

  c = x² +25x +40

  9390 = x² +25x + 40

  x² +25x -9350 = 0 . . . . . . rearrange to standard form

  (x +110)(x -85) = 0 . . . . . . . factor the quadratic

The positive solution is the value of x that makes a factor be zero: x = 85.

The number of units manufactured at a cost of 9390 is 85 units.

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There are several ways you can solve the quadratic. There appear to be no restrictions here, so we used a graphing calculator to assist. It shows the zeros of the equation to be -110 and +85, so the factors are as shown above.

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The prime factorization of 9350 is ...

  9350 = 2 · 5² · 11 · 17

so there are 24 divisors:

  1, 2, 5, 10, 11, 17, 22, 25, 34, 50, 55, 85, 110, 170, 187, 275, 374, 425, 550, 850, 935, 1870, 4675, 9350

We're looking for two factors that differ by a relatively small amount, so we would start with the factors near the middle of this list:

  9350 = 85·110 . . . . . these differ by 25, so are the factors of interest

These let you rewrite the 25x term so you can factor by grouping:

  x² +110x -85x +9350 = 0 . . . . rewrite the middle term

  x(x +110) -85(x +110) = 0 . . . . . factor each pair of terms

  (x -85)(x +110) = 0 . . . . . . . . . . factor out the common factor

The solution of interest is x = 85.

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You can also "complete the square"

  x² +25x = 9350

  x² +25x + 156.25 = 9506.25

  (x² +12.5) = 97.5² . . . . . rewrite as squares

  x = 97.5 -12.5 = 85 . . . . take the positive square root