Directions:
Read the following case studies carefully and answer the questions that follow.
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Case Study 1: Adventure Camp
A group of students went to an adventure camp. They stayed in a tent which is in the shape of a cylinder surmounted by a conical top. The height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m.
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Find the area of the canvas used for making the tent.
Solution: (A) 44 m$^2$
Step 1: Radius $r = 2$ m. Height of cylinder $h = 2.1$ m. Slant height $l = 2.8$ m.
Step 2: Area of canvas = CSA of Cylinder + CSA of Cone $= 2\pi rh + \pi rl = \pi r(2h + l)$.
Step 3: $= \frac{22}{7} \times 2 (2 \times 2.1 + 2.8) = \frac{44}{7} (4.2 + 2.8) = \frac{44}{7} \times 7 = 44$ m$^2$. -
Find the cost of the canvas of the tent at the rate of ₹ 500 per m$^2$.
Solution: (B) ₹ 22,000
Step 1: Cost = Area $\times$ Rate $= 44 \times 500 = 22000$.
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Find the area of the canvas used for making the tent.
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Case Study 2: Juice Stall
A juice seller was serving his customers using glasses. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of a glass was 10 cm. (Use $\pi = 3.14$)
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Find the apparent capacity of the glass.
Solution: (A) 196.25 cm$^3$
Step 1: Radius $r = 2.5$ cm. Height $h = 10$ cm.
Step 2: Apparent capacity = Volume of cylinder $= \pi r^2 h = 3.14 \times (2.5)^2 \times 10 = 3.14 \times 6.25 \times 10 = 196.25$ cm$^3$. -
Find the actual capacity of the glass.
Solution: (B) 163.54 cm$^3$
Step 1: Volume of hemisphere $= \frac{2}{3}\pi r^3 = \frac{2}{3} \times 3.14 \times (2.5)^3 = \frac{2}{3} \times 3.14 \times 15.625 \approx 32.71$ cm$^3$.
Step 2: Actual capacity = Apparent capacity - Volume of hemisphere $= 196.25 - 32.71 = 163.54$ cm$^3$.
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Find the apparent capacity of the glass.