Question

the park wants to put up a large sign outside the entrance gate to help visitors find it more easily.

• the sign will be made from metal and will be attached to two square based steel poles.
• the poles will be cemented into the ground into square based rectangular holes with a depth of 0,065 m.
• the volume of soil removed to make one hole is 0,012584 m³
• the amount of space in the hole that will be taken up by each pole is 0,0041 m³.

sketch of the pole cemented in the hole
sketch of square based pole
easy mix concrete to be used
price per container: r105
instructions: mix contents with 2 litres of water to make 10 litres of concrete.
you may use the following formulae:
surface area of pole = 2 × (side)² + 4 × (height × side)
volume of hole = (side of hole)² × depth
note:
1 m³ = 1 000 litre
adapted from: cashbuild, builders warehouse & department of transport
use the information above to answer the questions that follow:
4.2.1 the entire pole needs to be covered with a protective coating to make it last longer.
determine, in m², the total surface area of the pole that will need to be covered.
4.2.2 calculate, in m, rounded to two decimal places, the length of a side of the hole.
4.2.3 the construction company stated that the total cost of the concrete for the project will be r210.
verify, showing all calculations, whether this statement is valid.

Explanation:

4.2.1

Step1: Convert side length to meters

Given side length of pole \(25\ cm\). Since \(1\ m = 100\ cm\), then \(25\ cm=\frac{25}{100}=0.25\ m\)

Step2: Substitute into surface - area formula

The formula for the surface area of the pole \(S = 2\times(side)^{2}+4\times(height\times side)\). Here, \(height = 2.4\ m\) and \(side = 0.25\ m\)

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Step1: Use the volume formula for the hole

The volume formula for the hole \(V=(side\ of\ hole)^{2}\times depth\). We know \(V = 0.012584\ m^{3}\) and \(depth=0.065\ m\)
Let the side of the hole be \(x\). Then the equation is \(0.012584=x^{2}\times0.065\)

Step2: Solve for \(x\)

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Step1: Calculate the volume of concrete per hole

The volume of concrete per hole \(V_{concrete\ per\ hole}=Volume\ of\ hole - Volume\ of\ pole\). Given \(V_{hole}=0.012584\ m^{3}\) and \(V_{pole}=0.0041\ m^{3}\), so \(V_{concrete\ per\ hole}=0.012584 - 0.0041=0.008484\ m^{3}\)

Step2: Calculate the total volume of concrete for two holes

\(V_{total}=2\times0.008484 = 0.016968\ m^{3}\)
Since \(1\ m^{3}=1000\) litres, \(V_{total}=0.016968\times1000 = 16.968\) litres

Step3: Calculate the number of containers

Each container makes \(10\) litres of concrete. The number of containers \(n=\frac{16.968}{10}=1.6968\approx2\) (since we can't buy a fraction of a container)

Step4: Calculate the total cost

The cost per container is \(R105\). The total cost \(C = 2\times105=R210\)

Answer:

\(2.525\ m^{2}\)

4.2.2