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 $16 to produce 2 widgets, $18 to produce 4 widgets, and $48 to produce 10 widgets. find the total cost of producing 6 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 $16 to produce 2 widgets, $18 to produce 4 widgets, and $48 to produce 10 widgets. find the total cost of producing 6 widgets. enter the correct answer.

Answer

Explanation:

Step1: Assume the quadratic function

Let $c(x)=ax^{2}+bx + c$.

Step2: Substitute given - points

When $x = 2$, $c(2)=4a + 2b + c=16$; when $x = 4$, $c(4)=16a+4b + c = 18$; when $x = 10$, $c(10)=100a+10b + c = 48$.

Step3: Solve the system of equations

Subtract the first equation from the second: $(16a + 4b + c)-(4a + 2b + c)=18 - 16$, which simplifies to $12a+2b = 2$, or $6a + b=1$. Subtract the second equation from the third: $(100a+10b + c)-(16a + 4b + c)=48 - 18$, which simplifies to $84a+6b = 30$, or $14a + b = 5$. Then subtract the new - first equation ($6a + b=1$) from the new - second equation ($14a + b = 5$): $(14a + b)-(6a + b)=5 - 1$, $8a=4$, so $a=\frac{1}{2}$. Substitute $a=\frac{1}{2}$ into $6a + b = 1$, we get $6\times\frac{1}{2}+b = 1$, $3 + b = 1$, $b=-2$. Substitute $a=\frac{1}{2}$ and $b = - 2$ into $4a + 2b + c=16$: $4\times\frac{1}{2}+2\times(-2)+c=16$, $2-4 + c=16$, $c = 18$. So $c(x)=\frac{1}{2}x^{2}-2x + 18$.

Step4: Find the cost of 6 widgets

Substitute $x = 6$ into $c(x)$: $c(6)=\frac{1}{2}\times6^{2}-2\times6 + 18$. $c(6)=\frac{1}{2}\times36-12 + 18$. $c(6)=18-12 + 18$. $c(6)=24$.

Answer:

24