2.A poster is to contain 108 cm^2 of printed matter with margins of 6cm each at top and bottom and 2 cm on the sides. What is the minimum cost of the of the poster if it is to be made of material costing 20 cents/cm^2? ans. is $60 2. 1. So$\displaystyle V = x^{2}y $which we want to maximize. Then$\displaystyle 4x+y \leq 108 $. Note that girth is$\displaystyle 2(h+w) $or in this case$\displaystyle 2(x+x) = 4x $. To get the maximum volume we set$\displaystyle 4x+y = 108 $. Then$\displaystyle y = 108 - 4x $and$\displaystyle 0 \leq x \leq 27 $. Then$\displaystyle V = x^2(108-4x) $.$\displaystyle V'(x) = 216x-12x^2 $.$\displaystyle 0 = 216x-12x^2 = 12x(18-x) $. Critical values are$\displaystyle x = 0, \ x = 18 $. Checking the endpoints$\displaystyle 0,27 $(volume is 0) we conclude that the maximum volume occurs when$\displaystyle x = 18 $. So$\displaystyle V_{\text{max}} = 11,644 \ \text{in}^3 $. 2. So we want to minimize$\displaystyle C = 20(x-4)(y-12) $. Since$\displaystyle xy = 108 $,$\displaystyle y = \frac{108}{x} $. Then$\displaystyle C(x) = 20(x-4)\left(\frac{108}{x}-12 \right) $. So$\displaystyle C'(x) = \frac{2160}{x} - 240 - \frac{2160(x-4)}{x^2} $. Critical values are$\displaystyle 6,-6 $. We get$60 when we plug in $\displaystyle C(-6)$.