Question 1128496: I am beyond frustrated and failing to solve this, can anyone help please?
A company has determined that when x hundred dulcimers are built, the average cost per dulcimer can be estimated by C(x)=0.3x^2−2.7x+7.475, 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 $____.
The company should build ____dulcimers to achieve the minimum.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
For the general quadratic equation
f(x) = ax^2 + bx + c
the vertex point is (h,k) where h = -b/(2*a) and k = f(h)
The min cost will occur at the vertex as the vertex is the lowest point when a > 0
Based on that general form above, we see that
C(x) = 0.3x^2 - 2.7x + 7.475
leads to these three coefficients:
a = 0.3
b = -2.7
c = 7.475
Plug in the values of 'a' and 'b' to find h
h = -b/(2*a)
h = -(-2.7)/(2*0.3)
h = 2.7/0.6
h = 4.5
This is the x coordinate of the vertex
Now plug in x = 4.5 to find the value of k
C(x) = 0.3x^2 - 2.7x + 7.475
C(4.5) = 0.3(4.5)^2 - 2.7(4.5) + 7.475
C(4.5) = 0.3(20.25) - 2.7(4.5) + 7.475
C(4.5) = 6.075 - 12.15 + 7.475
C(4.5) = 1.4
k = C(h) = C(4.5) = 1.4
Therefore, the vertex point is (h,k) = (4.5, 1.4)
The minimum average cost per dulcimer is $__1.40__.
The company should build __450__dulcimers to achieve the minimum.
note: x is the number of dulcimers built and this number is in hundreds. So x = 4.5 indicates 4.5 hundred or 4.5*100 = 450 units are built.
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