A solid sphere of radius 3 cm is placed inside a cylinder and then water is poured into the cylinder until it is full, as shown in the diagram - Leaving Cert Mathematics - Question 8 - 2019

The volume of water displaced by the sphere is equal to the volume of the sphere, which is ( 36\pi \text{ cm}^3 ). The internal radius of the cylinder is 5 cm, so the height of water h can be calculated using the formula for the volume of a cylinder: $V = \pi r^2 h$ Substituting the volume and radius: $36\pi = \pi (5^2) h$ $36 = 25h$ Solving for h yields: $h = \frac{36}{25} = 1.44 \text{ cm}$ Therefore, the drop in the height of the water is ( 1.44 \text{ cm} ).

The curved surface area (C.S.A) of the cylinder is given by: $C.S.A = 2\pi rh$ where r = 5 cm and h = 18 cm: $C.S.A = 2\pi(5)(18) = 180\pi ext{ cm}^2$

The area of the rectangular piece of metal is: $A_{rectangle} = 35 cm \times 20 cm = 700 cm^2$

The amount of metal left over is: $A_{remaining} = A_{rectangle} - C.S.A = 700 - 180\pi$ Using ( \pi \approx 3.14 ): $A_{remaining} = 700 - 565.49 = 134.51\text{ cm}^2$ Rounded to 1 decimal place, the amount of metal left over is ( 134.5 ext{ cm}^2 ).

The margin of error (E) for a sample size (n=800) can be calculated using the formula: $E = \frac{1}{\sqrt{n}}$ Substituting in the sample size: $E = \frac{1}{\sqrt{800}} \approx 0.0355$ This gives the margin of error as a percentage: $E_{percentage} = 0.0355 \times 100 \approx 3.54\%$ Thus, the margin of error is ( 3.54% ).

The sample proportion ( \hat{p} ) is: $\hat{p} = \frac{350}{800} = 0.4375 \text{ or } 43.75\%$ Using the margin of error calculated: Confidence interval = ( \hat{p} \pm E ) Hence: $43.75 - 3.54 < p < 43.75 + 3.54$ This results in: $40.21 < p < 47.29$ Thus, the 95% confidence interval is (40.21%, 47.29%).

Conclusion: The newspaper's claim that the level of support is 50% is not substantiated by this survey. Reason: The 95% confidence interval (40.21%, 47.29%) does not include 50%, indicating that we do not have sufficient evidence to support the claim that 50% of people in the area support the EU proposal.