An open box folded from a sheet of…

Name: ______________________ Date: ____________ An open box folded from a sheet of cardboard (6 points)

Table

A packaging designer makes an open box from a rectangular sheet of cardboard

30

centimeters long and

20

centimeters wide. She cuts a square with sides of

x

centimeters from each corner and folds up the sides. The table gives the volume

V(x)

of the box, in cubic centimeters, for selected values of

x

Volume V(x) of the open box, in cubic centimeters, for cut size x, in centimeters
$x$ (centimeters)	1	2	3
$V(x)$ (cubic centimeters)	504	832	1008

(A(i)) Write an expression for $V(x)$ , the volume of the box, in terms of $x$ . [1 point]
(A(ii)) Use your expression to find the cut size $x$ that gives the greatest volume, and the greatest volume. Give your answers correct to three decimal places. [1 point]
(B(i)) Find the average rate of change of $V$ on the interval $[1, 3]$ , with units. Explain what this value represents on the graph of $V$ . [1 point]
(B(ii)) Use the secant line from part B(i) to estimate $V(2)$ . [1 point]
(C(i)) Is the estimate in part B(ii) an overestimate or an underestimate of $V(2)$ ? Give a reason for your answer. [1 point]
(C(ii)) The designer uses the model $V$ only for values of $x$ that make sense in this context. Explain what the largest possible value of $x$ is, and why the model cannot be used past it. [1 point]

An open box folded from a sheet of cardboard