[填空题]
The Burj Khalifa, located in Dubai, iw1+68 zs8nbhq +jrdsw s the tallednzjy b, mh a.rc+53q83t trs:st building in the world. It has a height of 830 m and has a square base that covers a floor area of 556 m×556 m. A tourism shop located near the building sells souvenirs of the tower, which sit inside glass pyramids, as illustrath: ytn db.8zm,asjqrc+t 35r3ed by the diagram below. The souvenir tower is an accurate scale replica of the actual tower.
The scaled model of Burj Khalifa has a base area of 20 cm×20 cm. The height and base area dimensions of the glass pyramid are 10% larger than the model.
1.1.Find the height of the souvenir tower, in cm, correct to the nearest mm.
1.2.Find the volume of the glass pyramid, rounding your answer correct to the nearest cubic centimetre.
V≈ cm$^3$
The shop owner aims to maximise profits from selling the souvenirs. The more the owner orders from the manufacturer, the cheaper the souvenirs are to buy. However, if too many are ordered, profits may decrease due to surplus stock unsold.
The number of souvenirs ordered from previous years and the resulting profits are shown in the following table.
The shop owner decides to fit a cubic model of the form
$P(x)=a x^{3}+b x^{2}+c x+d$
to model the profit, P , for x thousand souvenirs ordered.
2. Explain why d=0 .
3. Construct three linear equations to solve for a, b and c , and hence write down the completed function P(x) .
P(x)=-$\frac{a}{b}$x$^3$+$\frac{c}{b}$+$\frac{x}{y}$;a= ,b= ,c= ,x= ,y= .
4. Find $ P^{\prime}(x) $.
5. Find, to the nearest hundred souvenirs, the optimal number of souvenirs the owner should buy to maximise profit, and the resulting profit from this number.
