[填空题]
Jeremy is making an open-top rectangular box as part of u;hir hzppr80)rc;c1mx,+a ya science project. Hoggbd9q) gw 0)uj;ru6 qr,xp8e makes the box from a rectangular piece of cardboard, 30 cm x 18 cm. To construct the box, Jeremy cuts off squares of side length x c bj ru,8;6wq9gx r)qd)ug0pgom 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.
