A company has determined that when x hundred dulcimers are built, the average cost per dulcimer can be estimated by C(x)=0.2x(squared)−2.6x+9.950, where C(x) is in hundreds of dollars. What is the minimum average cost per dulcimer and how many dulcimers should be built to achieve that minimum?

Question

A company has determined that when x hundred dulcimers are built, the average cost per dulcimer can be estimated by C(x)=0.2x(squared)−2.6x+9.950, where C(x) is in hundreds of dollars. What is the minimum average cost per dulcimer and how many dulcimers should be built to achieve that minimum?

The minimum average cost per dulcimer is $______. I have gotten this wrong twice and I only have one more try tonight :(

Answer 1

the graph of a quadratic equation is a parabola

when you are looking for a minimum, the parabola generally opens upward, with a minimum at the vertex
... this means a negative coefficient for the 2nd order (squared) term
... this is not the case with the equation you have

there seems to be something amiss

Answer 2

This is the question word for word on MyMathLab. The title of the chapter it is from is "Problem Solving and Quadratic Functions"

Answer 3

sorry, my train of thought derailed ...

the vertex is on the axis of symmetry
... the equation (for the axis) is ... x = -b / 2a
... in this case ... x = -(-2.6) / (2 * .2) = 6.5

min # of dulcimers is ... 650

solve C(650) to find the min ave cost of dulcimers

Answer 4

To find the minimum average cost per dulcimer and the number of dulcimers needed to achieve this minimum cost, we can use calculus.

Step 1: Calculate the derivative of the cost function C(x).
The derivative of C(x) = 0.2x^2 - 2.6x + 9.950 with respect to x can be found by taking the derivative of each term separately:

dC(x)/dx = (d/dx)(0.2x^2) - (d/dx)(2.6x) + (d/dx)(9.950)
= 0.2(2x) - 2.6 + 0
= 0.4x - 2.6

Step 2: Set the derivative equal to zero and solve for x.
To find the minimum average cost, we need to find the critical points where the derivative is equal to zero:

0.4x - 2.6 = 0

Solving for x, we get:
0.4x = 2.6
x = 2.6 / 0.4
x = 6.5

So, 6.5 hundred dulcimers should be built to achieve the minimum average cost.

Step 3: Substitute the value of x back into the cost function C(x) to find the minimum average cost.
Substituting x = 6.5 into C(x) = 0.2x^2 - 2.6x + 9.950, we get:

C(6.5) = 0.2(6.5)^2 - 2.6(6.5) + 9.950
= 0.2(42.25) - 16.9 + 9.950
= 8.45 - 16.9 + 9.950
= 1.5

Therefore, the minimum average cost per dulcimer is $1.50.

I hope this helps you get the correct answer!