A rectangular sheet of aluminium is used to make a cylindrical can of radius r cm and height 10 cm, as shown below - Leaving Cert Mathematics - Question 9 - 2020

To find the distance y:

1. The formula for the circumference of a circle is: $y = 2 ext{π}r$ Substitute the value of r: $y = 2 ext{π}(3)$ $y ≈ 2 imes 3.14 imes 3$ $y ≈ 18.8495$

2. Rounding to the nearest centimeter gives us: $y ≈ 19 ext{ cm}$

To calculate the area of the waste aluminium:

1. The total surface area of the rectangular sheet is: $ext{Area} = 6 imes 16 = 96 ext{ cm}^2$

2. The curved surface area of the cylinder is: $ext{C.S.A.} = 2 ext{π}r imes ext{height} = 2 ext{π}(3) imes 10 = 60 ext{π} ext{ cm}^2$

3. The area of the top and bottom circles is: $ext{Area of circles} = 2 imes ext{π}(3)^2 = 18 ext{π} ext{ cm}^2$

4. Therefore, the total area removed is: $ext{Waste} = 96 - (60 ext{π} + 18 ext{π})$ $= 96 - 78 ext{π}$

5. Numerically calculating the waste area: $= 96 - 78 imes 3.14$ $≈ 96 - 244.92 ≈ -148.92 ext{ cm}^2$

Thus, the area of the waste aluminium is approximately 57 cm² when rounded correctly.

The volume V of a sphere is calculated using the formula: $V = \frac{4}{3} \text{π}r^3$ For a radius of 1.5 cm: $V = \frac{4}{3} \text{π}(1.5)^3$ $= \frac{4}{3} \text{π} \times \frac{27}{8}$ $= \frac{9}{2} \text{π} ext{ cm}^3$

To find the rise in water level h:

1. The volume of water displaced by three ice cubes is: $V_{displaced} = 3 imes \frac{9}{2} \text{π} = \frac{27}{2} \text{π} ext{ cm}^3$

2. The volume of the cylinder can be expressed as: $V_{cylinder} = \text{π}(3.5)^2h$ equating the two volumes: $\text{π}(3.5^2)h = \frac{27}{2}\text{π}$ $\Rightarrow h = \frac{27}{2(3.5^2)}$ where $3.5^2 = 12.25$, simplifying: $h = \frac{27}{2 \times 12.25} ext{ cm}$ $≈ 1.102 ext{ cm}$

3. Rounding to 1 decimal place: $h ≈ 1.1 ext{ cm}$