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Two cylinders, A and B, are given. The radius of cylinder B is twice that of cylinder A, and the height of cylinder B is half that of cylinder A. Find the ratio of the curved surface area of A to the curved surface area of B. Also find the ratio of the volume of A to the volume of B.
Answer: Let cylinder A have radius $r$ and height $h$. Cylinder B has radius $2r$ and height $h/2$.
Curved Surface Area (CSA):
$CSA_A = 2\pi rh$
$CSA_B = 2\pi (2r)(h/2) = 2\pi rh$
Ratio of CSA = $1 : 1$.
Volume (V):
$V_A = \pi r^2 h$
$V_B = \pi (2r)^2 (h/2) = \pi (4r^2)(h/2) = 2\pi r^2 h$
Ratio of Volume = $1 : 2$.
The edge of a cube measures $r$ cm. The largest possible right circular cylinder is cut out of the cube. What do you think is the volume of the cylinder (in cm³)?
Answer: The largest cylinder that fits in a cube of edge $r$ will have a diameter equal to the edge length ($2R = r \Rightarrow R = r/2$) and a height equal to the edge length ($h = r$).
Volume of cylinder = $\pi R^2 h = \pi \left(\frac{r}{2}\right)^2 (r) = \pi \left(\frac{r^2}{4}\right) (r) = \frac{\pi r^3}{4}$ cm³.
The radius of the base of a cylinder is increased by 10%. At the same time, the height of the cylinder is decreased by x%. Given that the volume of the cylinder remains unchanged, find the value of x.
Answer: Let the original radius be $r$ and height be $h$. Volume $V = \pi r^2 h$.
New radius = $1.1r$. New height = $h \left(1 - \frac{x}{100}\right)$.
New Volume = $\pi (1.1r)^2 \left(h \left(1 - \frac{x}{100}\right)\right) = 1.21 \pi r^2 h \left(1 - \frac{x}{100}\right)$.
Since the volume is unchanged: $1 = 1.21 \left(1 - \frac{x}{100}\right)$.
$1 - \frac{x}{100} = \frac{1}{1.21} = \frac{100}{121}$
$\frac{x}{100} = 1 - \frac{100}{121} = \frac{21}{121}$
$x = \frac{2100}{121} \approx 17.36\%$.
A solid metallic cube of side 12 cm is melted and recast into solid cylindrical rods, each having radius 2 cm and height 12 cm. Find:
(i) the volume of the cube,
(ii) the volume of one cylindrical rod,
(iii) the approximate number of complete cylindrical rods that can be formed. ($\pi \approx \frac{22}{7}$)
Answer:
(i) Volume of the cube = $a^3 = 12^3 = 1728$ cm³.
(ii) Volume of one cylindrical rod = $\pi r^2 h = \frac{22}{7} \times 2^2 \times 12 = \frac{22}{7} \times 48 = \frac{1056}{7} \approx 150.85$ cm³.
(iii) Number of complete rods = $\frac{\text{Volume of cube}}{\text{Volume of one rod}} = \frac{1728}{150.85} \approx 11.45$. So, 11 complete cylindrical rods can be formed.
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