2 FoM026 Notes O Ask Your Teac A graphing calculator is reco

2 FoM.026. Notes O Ask Your Teac A graphing calculator is recommended In this problem you are asked to find a function that models a real-life situation and then use the model to answer questions about the situation. Use the guildellnes on page 237 to help you A box with an open top ls to be constructed from a rectangular plece of cardboard with dimensions 12 in. by 30 in. by cutting out equal squares of side x at each corner and then folding up the sides (see the figure). 30 in. 12 in. (a) Find a function that models the volume V of the box. (b) Find the values of x for which the volume is greater than 250 in3. (Round your answers to three decimal places. Enter your answer using interval notation.) (c) Find the largest volume that such a box can have. (Round your answer to three decimal places.)

Solution

Volume of the box (cuboid in shape)

= length ×breadth ×height

Length= (30-2x)inch

Breadth = (12-2x)

Height = x (given)

Thus the volume function

= (30-2x)(12-2x)x= 360x-60x²-24x²+4x³3

= (4x³-84x²+360x )inch³

4x³-84x²+360x>250 take any value larger than 250 to make calculations easy.

x³-21x²+90x=280/4= 70

x³-21x²+90x-70=0 for x=1 the equation becomes zero thus for x=1 the volume is larger than 250inch³