3.1 The Big Five Marathon is an annual event in South Africa - NSC Mathematical Literacy - Question 3 - 2018 - Paper 2

Upon reaching the 20 km mark, a marathon runner is heading South East (S.E) or South East direction.

The runner's statement is deemed correct as he was experiencing a higher altitude compared to sea level. Thus, the highest point reached surpasses standard sea level measurements.

Cut-off times ensure that all runners complete the race within a reasonable period, promoting safety and allowing for the proper management of resources and support services along the route.

We first establish the time taken by the half-marathon runner:

• Required speed for half-marathon: $\text{Speed (half-marathon)} = \frac{16.5 \text{ km}}{5 \text{ hours}} = 3.3 \text{ km/h}$

Next for the full marathon:

• Runner’s speed comparison: $\text{Speed (full marathon)} = 3.3 + 2.7 = 6 \text{ km/h}$

We now calculate the time required for the full marathon to meet cut-off:

• Time to cover 31.5 km: $\text{Time} = \frac{31.5 \text{ km}}{6 \text{ km/h}} = 5.25 \text{ hours}$

This is 15 minutes short of the cut-off time of 5 hours 15 minutes, thus validating the runner’s claim to be correct.

First, we calculate the inner diameter by subtracting the thickness of the walls:

• Outer diameter = 31.2 cm
• Wall thickness = 0.2 cm on each side, so: $\text{Inner diameter} = 31.2 - 2(0.2) = 30.8 \text{ cm}$

Next, we convert the radius into centimeters:

• Radius = 30.8/2 = 15.4 cm.

Using the volume formula, where volume = 20 liters or 20000 cm³: $20000 = \pi(15.4)^2h$

Now, solving for height $h$: $h = \frac{20000}{3.142 \times (15.4)^2} = 26.84 \text{ cm}$

To determine the area of the solid pallet, we calculate the total area covered by the buckets:

• Area of one bucket: $A = \pi(15.4)^2 = 749.79 \text{ cm}^2$

For 11 buckets: $Total = 11 \times 749.79 = 8247.69 \text{ cm}^2$

Now, for the rectangular floor:

• Length of the pallet = 120 cm; Width = 120 cm:
• Total area of pallet = 14400 cm²

Unused area: $Unused = Total\,Area\,of\,Pallet - Total\,Area\,Buckets = 14400 - 8247.69 = 6152.31 \text{ cm}^2$

Given that the total length of the pallet is 120 cm and there are 12 buckets:

• Each bucket occupies 31.2 cm in diameter. Thus: $C = 120 - (4 \times 31.2) = 120 - 124.8 = -4.8\text{ cm}$

This implies that the pallet length might be insufficient.

To accommodate 12 buckets:

• Required length for buckets: $C = 12 \times 31.2 = 374.4\text{ cm}$

The extra length required: $Extra = C - Length = 374.4 - 120 = 254.4\text{ cm}$

Then, the percentage increase is calculated as follows: $Percentage\,Increase = \frac{Extra}{Length} \times 100 = \frac{254.4}{120} \times 100 = 212.0\%$\