17 A – Cubic Polynomials 4: Modeling with Cubics

Volume A 40 cm by 30 cm sheet of tinplate is to be used to make a cake tin. Squares are cut from its corners and the metal is then folded upwards along the dashed lines. Edges are fixed together to form the open rectangular tin. The capacity (volume) of the cake dish is V = lwh or V(x) = (40 – 2x)(30 – 2x)(x) where (40 – 2x) is the length, (30 – 2x) is the width, and (x) is the height.  How does the capacity change as x changes?

Modeling With Cubics  What are the restrictions on x? – x must be positive (represents height) – Since each of these quantities must be greater than 0, we know that x > 15. – When we graph this cubic, we can limit the window for x values between 0 and 15.  What value of x produces the cake tin of maximum capacity?