Q:

The cost to produce a product is modeled by the function f(x) = 5x2 − 70x + 258 where x is the number of products produced. Complete the square to determine the minimum cost of producing this product. 5(x − 7)2 + 13; The minimum cost to produce the product is $13. 5(x − 7)2 + 13; The minimum cost to produce the product is $7. 5(x − 7)2 + 258; The minimum cost to produce the product is $7. 5(x − 7)2 + 258; The minimum cost to produce the product is $258.

Accepted Solution

A:
The cost to produce a product is modeled by the function [tex] f(x)=5x^{2}-70x+258 [/tex] , where x is the number of products produced.We have to determine the minimum cost of producing this product.Since, [tex] f(x)=5x^{2}-70x+258 [/tex]Now, consider the equation [tex] 5x^{2}-70x+258=0 [/tex]Dividing the above equation by 5, we get[tex] x^{2}-\frac{70x}{5}+\frac{258}{5}=0 [/tex][tex] x^{2}-14x+\frac{258}{5}=0 [/tex]Now, considering the coefficient of 'x', dividing it by '2' and then adding and subtracting the square of the number which we got after dividing .Since, coefficient of 'x' is 14, and half of 14 is '7'.So, adding and subtracting [tex] 7^{2} [/tex] from the above equation.[tex] x^{2}-14x+(7)^{2}-(7)^{2}+\frac{258}{5}=0 [/tex][tex] x^{2}-14x+49-49+\frac{258}{5}=0 [/tex][tex] (x-7)^{2}-49+\frac{258}{5}=0 [/tex][tex] (x-7)^{2}+\frac{258-245}{5}=0 [/tex][tex] (x-7)^{2}+\frac{13}{5}=0 [/tex][tex] 5(x-7)^{2}+13=0 [/tex]Now, we have to determine the minimum cost to produce the product.Since, [tex] f(x)=5x^{2}-70x+258 [/tex][tex] f'(x)=10x-70 [/tex]Now, let f'(x)=0[tex] 10x-70=0 [/tex][tex] 10x=70 [/tex]Therefore, x=7Now, consider [tex] f''(x)=10 [/tex] which is greater than 0.Therefore, x= 7 is the minimum cost.The minimum cost to produce the product is $7.