Practice Problems (Junior Cert Mathematics): Revision Notes

Practice Problems

Problem:

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Solutions:

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To find the volume of the rectangular block of wax, we use the formula for the volume of a rectangular prism:

$\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}$

1. Step 1: Identify the dimensions of the block:

• Length = 35 cm (This is how long the block is.)
• Width = 45 cm (This is how wide the block is.)
• Height = 16 cm (This is how tall the block is.) Explanation: These are the three dimensions needed to calculate the volume. They tell us how much space the block takes up.

2. Step 2: Plug in the values into the formula and calculate the volume:

• Volume = 35 × 45 × 16 = 25,200 cubic centimetres Explanation: We multiply the length, width, and height together in one step to find the total volume of the block.

Final Answer:

The volume of the block of wax is 25,200 cm³.

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Next, we need to find the volume of each cylindrical candle. We use the formula for the volume of a cylinder:

$\text{Volume of a cylinder} = \pi r^2 h$

3. Step 1: Identify the dimensions of the candle:

• Radius = r cm (This is half the diameter of the base of the cylinder.)
• Height = 9 cm (This is how tall the candle is.) Explanation: The radius and height are the key dimensions we need to calculate the volume of the cylinder.

4. Step 2: Plug in the height and leave the radius as r since the problem asks for the answer in terms of r and π:

• Volume = π r² × 9 = 9π r² Explanation: We substitute the height (9 cm) into the formula, and leave r and π as they are because the question asks for the answer in these terms.

Final Answer:

The volume of each candle is 9π r² cm³.

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Finally, we need to calculate the radius r of each candle, given that the students make 100 candles from the block of wax and 10% of the wax is wasted.

5. Step 1: Calculate the total usable volume of wax after waste:

• Total Volume = 25,200 cm³ (This is the volume of the block of wax from part (a).)
• Wasted Wax = 0.1 × 25,200 = 2,520 cm³
• Usable Wax = 25,200 - 2,520 = 22,680 cm³ Explanation: First, we calculate how much wax is wasted (10% of the total volume). Then, we subtract the wasted wax from the total volume to find out how much wax is actually available for making candles.

6. Step 2: Determine the total volume of 100 candles:

• Volume of 100 candles = Usable Wax = 22,680 cm³ Explanation: The total volume of all the candles combined must equal the total usable wax after waste.

7. Step 3: Use the volume formula from part (b) to set up the equation:

• 100 × 9π r² = 22,680 Explanation: Since each candle has a volume of 9π r², the volume of 100 candles is 100 × 9π r². This must equal the total usable wax volume (22,680 cm³).

8. Step 4: Simplify the equation to solve for r²:

• 900π r² = 22,680
• r² = 22,680/(900π) Explanation: We divide both sides of the equation by 900π to isolate r².

9. Step 5: Calculate r and correct to 1 decimal place:

• r² = 22,680/(900π) ≈ 22,680/2827.43 ≈ 8.02
• r ≈ √8.02 ≈ 2.8 cm Explanation: We calculate r² by dividing 22,680 by 900π. Then, we take the square root to find r. The final answer is rounded to 1 decimal place as required.

Final Answer:

The radius r of each candle is 2.8 cm.

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