Answer:15
Step-by-step explanation:
Given
Craftsman sell 10 Jewelry set for $500 each
For each additional set he will decrease the price by $ 25
Suppose he sells n set over 10 set
Earning[tex]=\text{Price of each set}\times \text{no of set}[/tex]
Earning [tex]=(500-25n)(10+n)[/tex]
[tex]E=5000+500n-25n^2-250n[/tex]
differentiate to get the maximum value
[tex]\frac{dE}{dn}=-50n+250[/tex]
Equate [tex]\frac{dE}{dn}[/tex] to get maximum value
[tex]-50n+250=0[/tex]
[tex]n=\frac{250}{50}[/tex]
[tex]n=5[/tex]
Thus must sell 5 extra set to maximize its earnings.
The craftsman should make 5 additional jewelry sets to maximize his earnings
The cost of 10 jewelry sets is given as: $500 per set
For every increment in the number of jewelry sets, the selling price decreases by $25
Let x represents the number of jewelry sets.
The cost function is:
[tex]\mathbf{C(x) = (500 - 25x) \times (10 + x)}[/tex]
Expand
[tex]\mathbf{C(x) = 5000 + 500x - 250x - 25x^2}[/tex]
[tex]\mathbf{C(x) = 5000 + 250x - 25x^2}[/tex]
Differentiate
[tex]\mathbf{C'(x) = 0+ 250 - 50x}[/tex]
[tex]\mathbf{C'(x) = 250 - 50x}[/tex]
Set to 0
[tex]\mathbf{250 - 50x = 0}[/tex]
Add 50x to both sides
[tex]\mathbf{50x = 250}[/tex]
Divide both sides by 50
[tex]\mathbf{x = 5}[/tex]
Hence, he should make 5 additional jewelry sets to maximize his earnings
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