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nataly862011 [7]
2 years ago
5

A company has determined that when x hundred dulcimers are​ built, the average cost per dulcimer can be estimated by ​C(x)=0.2x^

2-0.6x+2.250​, 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?
Mathematics
2 answers:
damaskus [11]2 years ago
8 0

Answer:

The minimum average cost per dulcimer is 150 and the minimum cost is $180.

Step-by-step explanation:

The given cost function is

C(x)=0.2x^2-0.6x+2.250

Differentiate the function C(x) with respect to x.

C'(x)=0.2(2x)-0.6(1)+(0)

C'(x)=0.4x-0.6

Equate C'(x)=0, to find the critical values.

C'(x)=0

0.4x-0.6=0

0.4x=0.6

Divide both sides by 0.4.

x=\frac{0.6}{0.4}

x=1.5

Differentiate the function C'(x) with respect to x.

C''(x)=0.4(1)

C''(x)=0.4

At x=1.5

C''(1.5)=0.4

The value of double derivative is positive. It means the function is minimum  at x=1.5 hundred.

1.5 hundred = 150

Substitute x=1.5 in the given function to find the minimum average cost.

C(1.5)=0.2(1.5)^2-0.6(1.5)+2.250

C(1.5)=1.8

The minimum cost is 1.8 hundred dollars.

1.8 hundred dollars = $180

Therefore, the minimum average cost per dulcimer is 150 and the minimum cost is $180.

liq [111]2 years ago
4 0

Answer: The minimum average cost per dulcimer = $ 180

The number of dulcimers should be built to achieve that​ minimum =150

                   

Step-by-step explanation:

Given : A company has determined that when x hundred dulcimers are​ built, the average cost per dulcimer can be estimated by

C(x)=0.2x^2-0.6x+2.250, where​ C(x) is in hundreds of dollars.

Now, differentiate the above function with respect to x, we get

C'(x)=0.4x-0.6                                        (1)

Put C'(x) =0, we get

0.4x-0.6=0\\\\\Rightarrow\ x=\dfrac{0.6}{0.4}=1.5

Again differentiate (1) w.r.t. x , we get

C"(x)=0.4>0        

By second derivative test , we have the value of x where C(x) is minimum=1.5

C(x)=0.2(1.5)^2-0.6(1.5)+2.250=1.8

Hence, the minimum average cost per dulcimer = $ 180

The number of dulcimers should be built to achieve that​ minimum =150

                   

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