It costs 14 dollars to manufacture and distribute a backpack. if the backpacks sell at x dollars each, the number sold, n, is given by n equals startfraction 2 over x minus 14 endfraction plus 5 (100 minus x ). find the selling price that will maximize profit.

Selling price that will maximize profit is $57

Explanation:

Given;

Costs to manufacture and distribute a backpack = $14

Number sold, n =

here, x is the selling cost of the bag

Now,

Profit = Total revenue - Total cost

or

P = nx - 14n

or

P = n(x - 14)

or

P =

or

P = 2 + 5(100 - x)(x - 14)

or

P = 2 + 5(100x - 1400 - x² + 14x)

differentiating with respect to x

we get

= 0 + 5(100 - 0 - 2x + 14)

or

= 5(114 - 2x)

put

= 0 for point of maxima or minima

5(114 - 2x) = 0

or

114 - 2x = 0

or

x = $57

Now,

= 5(0 - 2) = -10  

[hence, negative result means x = 57 is point of maxima]

Therefore,

Selling price that will maximize profit is $57

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