Question

solve quickly. a sheet of paper 19 cm by 8 cm is rolled into a cylinder so that its longer edges just meet. find: i) the diameter of the cylinder (to 2 decimal places) ii) the base of the cylinder (to 2 decimal places) iii) the curved - surface area of the cylinder (correct to the nearest whole) (take \\( \pi = 3.14 \\)) iv) the volume of the cylinder.

Explanation:

Step1: Determine the circumference and height of the cylinder

When the paper is rolled along its longer edges, the circumference of the base of the cylinder $C = 19$ cm and the height of the cylinder $h=8$ cm. Using the formula $C = 2\pi r$, we can find the radius $r$. So, $r=\frac{C}{2\pi}=\frac{19}{2\pi}$ cm.

Step2: Find the diameter of the cylinder

The diameter $d = 2r$. Substituting $r=\frac{19}{2\pi}$, we get $d=\frac{19}{\pi}\approx\frac{19}{3.14}\approx6.05$ cm.

Step3: Find the area of the base of the cylinder

The area of the base of a cylinder $A_{base}=\pi r^{2}$. Substituting $r = \frac{19}{2\pi}$, we have $A_{base}=\pi\times(\frac{19}{2\pi})^{2}=\frac{19^{2}}{4\pi}=\frac{361}{4\pi}\approx\frac{361}{4\times3.14}=\frac{361}{12.56}\approx28.75$ $cm^{2}$.

Step4: Find the curved - surface area of the cylinder

The curved - surface area of a cylinder $A_{curved}=Ch$. Since $C = 19$ cm and $h = 8$ cm, $A_{curved}=19\times8 = 152$ $cm^{2}$.

Step5: Find the volume of the cylinder

The volume of a cylinder $V=A_{base}h$. Substituting $A_{base}=\frac{361}{4\pi}$ and $h = 8$, we get $V=\frac{361}{4\pi}\times8=\frac{361\times2}{\pi}=\frac{722}{3.14}\approx229.94$ $cm^{3}$.

Answer:

i) Diameter $\approx6.05$ cm
ii) Area of the base $\approx28.75$ $cm^{2}$
iii) Curved - surface area = 152 $cm^{2}$
iv) Volume $\approx229.94$ $cm^{3}$