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You are given two options: (A) one chocolate cube of side 50 mm, (B) hundred chocolate cubes of side 10 mm. Which option gives you more chocolate?
Answer:
Option A Volume = $a^3 = 50^3 = 125,000$ mm³.
Option B Volume = $100 \times (10^3) = 100 \times 1000 = 100,000$ mm³.
Option A gives you more chocolate.
A school provides milk to its students in cylindrical glasses, each with a diameter of 7 cm. If each glass is filled with milk to a height of 12 cm, how many litres of milk are needed to serve 1600 students? Use 1000 cm³ = 1 litre.
Answer: Radius of the glass ($r$) = $7 / 2 = 3.5$ cm. Height ($h$) = 12 cm.
Volume of milk per student = $\pi r^2 h = \frac{22}{7} \times 3.5^2 \times 12 = \frac{22}{7} \times 12.25 \times 12 = 462$ cm³.
Total volume for 1600 students = $1600 \times 462 = 739,200$ cm³.
Volume in litres = $\frac{739,200}{1000} = 739.2$ litres.
The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone of radius 4 cm. Find the height and the volume of the cone.
Answer: Surface area of sphere ($r_s=5$) = $4\pi (5^2) = 100\pi$.
Curved surface area (CSA) of cone ($r_c=4$) = $\pi(4)l$.
Given: $100\pi = 5 \times (4\pi l) \Rightarrow 100\pi = 20\pi l \Rightarrow l = 5$ cm.
Height of cone ($h$) = $\sqrt{l^2 - r_c^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = 3$ cm.
Volume of the cone = $\frac{1}{3}\pi r_c^2 h = \frac{1}{3} \times \pi \times (4^2) \times 3 = 16\pi \approx 50.24$ cm³.
We have a cylinder with a base radius of r cm and height h cm. A square pyramid is fitted inside it. The square base of the pyramid lies on the base of the cylinder, its corners on the boundary of the cylinder. The apex of the pyramid lies on the top of the cylinder. Find the ratio of the volume of this pyramid to the volume of the cylinder.
Answer: Let the square base of the pyramid have side $a$. Its diagonal connects opposite corners on the circle's boundary, so the diagonal = $2r$.
By Pythagoras, $a^2 + a^2 = (2r)^2 \Rightarrow 2a^2 = 4r^2 \Rightarrow a^2 = 2r^2$. Thus, the area of the square base = $2r^2$.
The height of the pyramid is $h$.
Volume of the pyramid = $\frac{1}{3} \times \text{Area of base} \times \text{height} = \frac{1}{3}(2r^2)h = \frac{2}{3}r^2 h$.
Volume of the cylinder = $\pi r^2 h$.
Ratio = $\frac{\frac{2}{3}r^2 h}{\pi r^2 h} = \frac{2}{3\pi}$. Thus, the ratio is $2 : 3\pi$.
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