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?
Accepted Solution
A:
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 = 150Substitute 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 = $180Therefore, the minimum average cost per dulcimer is 150 and the minimum cost is $180.