Question

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.

Answer 1

The cost to produce a product is modeled by the function $f(x)=5x^{2}-70x+258$ , where x is the number of products produced.

We have to determine the minimum cost of producing this product.

Since, $f(x)=5x^{2}-70x+258$

Now, consider the equation $5x^{2}-70x+258=0$

Dividing the above equation by 5, we get

$x^{2}-\frac{70x}{5}+\frac{258}{5}=0$

$x^{2}-14x+\frac{258}{5}=0$

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 $7^{2}$ from the above equation.

$x^{2}-14x+(7)^{2}-(7)^{2}+\frac{258}{5}=0$

$x^{2}-14x+49-49+\frac{258}{5}=0$

$(x-7)^{2}-49+\frac{258}{5}=0$

$(x-7)^{2}+\frac{258-245}{5}=0$

$(x-7)^{2}+\frac{13}{5}=0$

$5(x-7)^{2}+13=0$

Now, we have to determine the minimum cost to produce the product.

Since, $f(x)=5x^{2}-70x+258$

$f'(x)=10x-70$

Now, let f'(x)=0

$10x-70=0$

$10x=70$

Therefore, x=7

Now, consider $f''(x)=10$ which is greater than 0.

Therefore, x= 7 is the minimum cost.

The minimum cost to produce the product is $7.

Answer 2

The minimum cost to produce the product is $7.

Given that

The cost to produce a product is modeled by the function;

$\rm f(x) = 5x^2 -70x + 258$

Where x is the number of products produced.

We have to determine

The minimum cost of producing this product.

According to the question

The cost to produce a product is modeled by the function;

$\rm f(x) = 5x^2 -70x + 258$

Dividing the above equation by 5,

$\rm x^2 -14x + \dfrac{258}{5}$

Adding and subtracting 49

$\rm x^2 -14x + \dfrac{258}{5} +49-49=0 \\ \\ \rm x^2 -14x+49 + \dfrac{258}{5} -49=0\\ \\ (x-7)^2 + \dfrac{258-245}{5}=0\\ \\ (x-7)^2 + \dfrac{13}{5}=0$

Multiply by 5 on both sides the equation

$\rm (x-7)^2+\dfrac{13}{5} =0\\ \\ 5(x-7)^2+13 = 0$

The minimum cost to produce the product is determined by the differentiation of the equation with respect to x.

$\rm 5\dfrac{d((x-7)^2)}{dx}+\dfrac{d(13 )}{dx}= 0\\ \\ 5 \times 2 (x-7) +0=0\\ \\ 10 (x-7) = 0\\ \\ 10x-70=0\\ \\ 10x=70\\ \\ x = \dfrac{70}{10}\\ \\ x=7$

Hence, The minimum cost to produce the product is $7.

To know more about Differentiation click the link given below.

https://brainly.com/question/24567634