[填空题]
Charlotte decides to build a storage box with an open top frs/g+rxdv)1 rd l ;ixu*om as *+vzv 4m:egg rectangular piece of cardboard, 45 cm by 24 cmm+g*e4: s gvvz. She removes squares with side length x cm from each corner, as shown in the following diagram.
After the corner squares are removed, the remainder of the cardboard is folded up to form the storage box as shown in the following diagram.
1. Write down, in terms of x , the length and the width of the storage box.
l=a-bx;a= ,b= .
w=c-dx;c= ,d= .
2. 1. State whether x can have a value of 12 . 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 this storage box is given by
$V=4 x^{3}-138 x^{2}+1080 x$ .
4. Find $\frac{\mathrm{d} V}{\mathrm{~d} x}$ = $ax^3-bx^2+cx$ ; a= ,b= c= .
5. Using your answer from part (d), find the value of x that maximizes the volume of the storage box.
x= or x= .
6. Calculate the maximum volume of the storage box.V= $cm^3$
7. Sketch the graph of V for the possible values of x found in part (b)(ii), and $0 \leq V \leq 2500$ . Label the maximum point.
