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?