SOLUTION: Mr. Lim orders chairs from a factory to sell. When he orders a batch of x chairs, the cost of each chair, $y is given by y = x2 - 14x + 80. a. Find x such that the cost of each

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Question 1165819: Mr. Lim orders chairs from a factory to sell. When he orders a batch of x chairs, the cost of each chair, $y is given by y = x2 - 14x + 80.
a. Find x such that the cost of each chair is the same as when Mr. Lim
ordered 5 chairs.
b. Find the number of chairs in a batch he needs to orders such that the cost
per chair is less than $45.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52784)   (Show Source): You can put this solution on YOUR website!
.
Mr. Lim orders chairs from a factory to sell. When he orders a batch of x chairs,
the cost of each chair, $y is given by y = x2 - 14x + 80.
(a) Find x such that the cost of each chair is the same as when Mr. Lim
ordered 5 chairs.
(b) Find the number of chairs in a batch he needs to orders such that the cost
per chair is less than $45.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

(a)  If he orders 5 chairs, the cost of each chair is

         5^2 - 14*5 + 80 = 35 dollars.


     To find x, we should solve this quadratic equation

         x^2 - 14x + 80 = 35.


     This equation is the same as

        x^2 - 14x + 80 - 35 = 0,

        x^2 - 14x + 45 = 0,

        (x-5)*(x-9) = 0     (after factoring).


     One root is  5 chairs  (the value we started with).

     The other root is 9, which is the ANSWER to question (a).



(b)  To answer (b), we should solve this inequality

        x^2 - 14x + 80 <= 45.


     Transform and simplify it

        x^2 - 14x + 80 - 45 <= 0,

        x^2 - 14x + 35 <= 0,

      
     Apply the quadratic formula to find the roots.

     The roots are   = .


     So, one root is   = 3.26 (approx.)  and  another root is   = 10.74 (approx.)


     Function x^2 - 14x + 35  is negative between the roots.


     Since we are interested to know integer values of x, they are  between 4 and 10 inclusive.


     So, the ANSWER to question (b)  is  "integer numbers between 4 and 10 inclusive".

Solved.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Here are alternative methods for solving each of the two parts of the problem.

The graph of the cost function is a parabola, with its minimum value when the number of chairs ordered is

a. Find x such that the cost of each chair is the same as when Mr. Lim
ordered 5 chairs.

Since the graph is symmetric about the axis of symmetry x=7, the cost will be the same for 9 chairs as it is for 5 chairs.

ANSWER: 9 chairs can be ordered for the same cost as 5 chairs

b. Find the number of chairs in a batch he needs to orders such that the cost
per chair is less than $45.

The minimum cost, when 7 chairs are ordered, is

Because the leading term of the cost function is , the cost will differ from the minimum cost of $31 by n^2 when x differs from 7 by n. We need the cost to be less than $45, which differs from the minimum cost of $31 by $14. Since 3^2 is less than 14 and 4^2 is greater than 14, the number of chairs that can be ordered for less than $45 can differ from 7 by at most 3.

ANSWER: Between 4 and 10 chairs can be ordered for a cost less than $45
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