The daily cost of producing x high performance wheels for racing is given by the following function, where no more than 100 wheels can be produced each day. What production level will give the lowest average cost per wheel? What is the minimum average cost?C(x)=0.09x^3 - 4.5x^2 + 180x; (0,100]
Accepted Solution
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Answer:The production of wheel per day is 74 which gives lowest average cost per wheel.The minimum average cost is $168.72.Step-by-step explanation:Given function of average cost is[tex]C(x)= 0.03x^3-4.5x^2+171x[/tex]Differentiating with respect to xC'(x)= 0.09 x² -9.0 x+171Again differentiating with respect to xC''(x) = 0.18 x -9.0To find the minimum average cost, first we have to set C'=0.The function's slope is zero at x=a, and the second derivative at x=a isless than 0, it is critical maximum.greater than 0, it is critical minimum.Now ,C'=0⇒ 0.09 x² -9.0 x+171=0⇒x = 74.49, 25.50[tex]C''(x)|_{x=74.49} = 0.18 (74.49)-9.0=4.41>0[/tex][tex]C''(x)|_{x=25.50} = 0.18 (25.50)-9.0=-4.41<0[/tex]Therefore at x= 74.49≈ 74, the average cost is minimum.The production of wheel per day is 74 which gives lowest average cost per wheel.The minimum average cost [tex]C(x)= 0.03x^3-4.5x^2+171x[/tex] [tex]=(0.03 \times 74^3)-(4.5 \times 74^2)+(171\times 74)[/tex] =168.72[Assume the average cost is in dollar]The minimum average cost is $168.72.