Respuesta :
Answer:
The minimum average cost per dulcimer is 150 and the minimum cost is $180.
Step-by-step explanation:
The given cost function is
[tex]C(x)=0.2x^2-0.6x+2.250[/tex]
Differentiate the function C(x) with respect to x.
[tex]C'(x)=0.2(2x)-0.6(1)+(0)[/tex]
[tex]C'(x)=0.4x-0.6[/tex]
Equate C'(x)=0, to find the critical values.
[tex]C'(x)=0[/tex]
[tex]0.4x-0.6=0[/tex]
[tex]0.4x=0.6[/tex]
Divide both sides by 0.4.
[tex]x=\frac{0.6}{0.4}[/tex]
[tex]x=1.5[/tex]
Differentiate the function C'(x) with respect to x.
[tex]C''(x)=0.4(1)[/tex]
[tex]C''(x)=0.4[/tex]
At x=1.5
[tex]C''(1.5)=0.4[/tex]
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.
[tex]C(1.5)=0.2(1.5)^2-0.6(1.5)+2.250[/tex]
[tex]C(1.5)=1.8[/tex]
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.
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
[tex]C(x)=0.2x^2-0.6x+2.250[/tex], where C(x) is in hundreds of dollars.
Now, differentiate the above function with respect to x, we get
[tex]C'(x)=0.4x-0.6[/tex] (1)
Put C'(x) =0, we get
[tex]0.4x-0.6=0\\\\\Rightarrow\ x=\dfrac{0.6}{0.4}=1.5[/tex]
Again differentiate (1) w.r.t. x , we get
[tex]C"(x)=0.4>0[/tex]
By second derivative test , we have the value of x where C(x) is minimum=1.5
[tex]C(x)=0.2(1.5)^2-0.6(1.5)+2.250=1.8[/tex]
Hence, the minimum average cost per dulcimer = $ 180
The number of dulcimers should be built to achieve that minimum =150
