[填空题]
Jeremy is making an openb h*g 0jc- qwz*rsv92vj(fh +h-top rectangulyob j3 ww,.jv -hdgxbc4x;f5*n2cu)l ar box as part of a science project. He makes the box from a rectangular piece of cardboard, 30 cm x 18 cm. To construct jl3 b,v)*g5cwb2h4nof ;y.-wdx jcuxthe box, Jeremy cuts off squares of side length x cm from each corner, as shown in the following diagram.
After removing the squares, Jeremy folds up the edges to form the box as shown.
1. Write down, in terms of x , expressions for the length and width of the box.
l=a-bx ;a= ,b= .
w=c-dx; c= ,d= .
2. 1. State whether x can have a value of 10 . Give a reason for your answer.
2. Write down the interval for the possible values of x .
3. Show that the volume, $V \mathrm{~cm}^{3}$ , of the box is given by
$V=4 x^{3}-96 x^{2}+540 x$ .
4. Find $\frac{\mathrm{d} V}{\mathrm{~d} x}$=ax^2-bx+c;a= ,b= ,c= .
5. Using your answer from part (d), find the value of x that maximizes the volume of the box.
x≈ cm.
6. Calculate the maximum volume of the box.V≈ $cm^3$.
7. Sketch the graph of V , for the possible values of x found in part (b)(ii), and $0 \leq V \leq 1000$ . Label the maximum point.
