widget wonders produces widgets. they have found that the cost, c(x), of making x widgets is a quadratic…

widget wonders produces widgets. they have found that the cost, c(x), of making x widgets is a quadratic function in terms of x. the company also discovered that it costs $15.50 to produce 3 widgets, $23.50 to produce 7 widgets, and $56 to produce 12 widgets. find the total cost of producing 5 widgets. enter the correct answer.

widget wonders produces widgets. they have found that the cost, c(x), of making x widgets is a quadratic function in terms of x. the company also discovered that it costs $15.50 to produce 3 widgets, $23.50 to produce 7 widgets, and $56 to produce 12 widgets. find the total cost of producing 5 widgets. enter the correct answer.

Answer

Answer:

18

Explanation:

Step1: Set up quadratic function

Let $c(x)=ax^{2}+bx + c$. We have the following system of equations based on the given data: When $x = 3$, $c(3)=9a + 3b + c=15.5$; when $x = 7$, $c(7)=49a+7b + c=23.5$; when $x = 12$, $c(12)=144a+12b + c=56$.

Step2: Subtract equations

Subtract the first - equation from the second: $(49a + 7b + c)-(9a + 3b + c)=23.5 - 15.5$, which simplifies to $40a+4b = 8$, or $10a + b = 2$, so $b = 2 - 10a$. Subtract the second - equation from the third: $(144a+12b + c)-(49a + 7b + c)=56 - 23.5$, which simplifies to $95a+5b = 32.5$, or $19a + b = 6.5$.

Step3: Solve for a

Substitute $b = 2 - 10a$ into $19a + b = 6.5$: $19a+(2 - 10a)=6.5$. $19a+2 - 10a=6.5$, $9a=4.5$, so $a = 0.5$.

Step4: Solve for b

Substitute $a = 0.5$ into $b = 2 - 10a$, we get $b=2-10\times0.5=-3$.

Step5: Solve for c

Substitute $a = 0.5$ and $b=-3$ into $9a + 3b + c=15.5$. $9\times0.5+3\times(-3)+c=15.5$, $4.5-9 + c=15.5$, $c=20$. So $c(x)=0.5x^{2}-3x + 20$.

Step6: Find cost for 5 widgets

When $x = 5$, $c(5)=0.5\times5^{2}-3\times5 + 20$. $c(5)=0.5\times25-15 + 20=12.5-15 + 20=18$.