[填空题]
For a homework assignrz l j6af7:bh.ment, Tiffdijxa xsk4+s)3om w4on;nd1* any makes a cylinder out of paper with a volume o)s mix +k a4;1*nsjxwdd3n o4of exactly 200 $\mathrm{~cm}^{3}$ . To make the curved surface of the cylinder, Tiffany cuts a rectangular shape from a standard A4 sheet of paper (A4 dimensions: 21 $\mathrm{~cm}$ width, 29.7 $\mathrm{~cm}$ length). The circumference of the cylinder base (the length of the rectangle cut) is denoted as x $\mathrm{~cm}$ and the height of the cylinder (the width of the rectangle cut) is denoted as y $\mathrm{~cm}$ .
1. Using the formula for the volume of a cylinder, find an expression of the width of the rectangular cut y , in terms of x .
y=$\frac{a{\pi}}{x^2}$;a= .
2. State whether x can have a value of 2 $\mathrm{~cm}$ . Give a reason for your answer.
y= cm
3. Show that the curved surface area of the cylinder can be expressed, in terms of x , as A=$\frac{800 \pi}{x} \mathrm{~cm}^{2}$ ,A=$\frac{a{\pi}}{x}$ $cm^2$ , a= .
4. Find $\frac{\mathrm{d} A}{\mathrm{~d} x}$ .
5. Calculate the value of x that minimises the curved surface area of the cylinder. Also find the corresponding value of y when this area is mini-mised.
6. Calculate the minimum curved surface area of the cylinder.
A≈ cm$^2$
