A civil engineering firm in Uganda is designing a cable-stayed bridge across River Nile to ease traffic flow. To plan resources, the project manager receives the following information: The durability (in years) of a new concrete mix that includes a special additive is modeled by: D(x)=28+2x-0.1x2 where x is the amount of additive in kg per cubic metre of concrete. The additive costs UGX 45,000 per kg. The concrete budget allows a maximum of UGX 450,000 per cubic metre for additives. Two bridge cable anchoring points are modelled in the complex plane as: A=3+4i, B=-2+7i . Coordinates are measured in tens of metres from a reference point on the riverbank. The firm schedules material delivery over five equal weekly instalments. The amount of additive delivered each week forms a geometric progression: 50,75,112.5,…(kg) . The firm has 5 mechanical engineers and 7 civil engineers. They need to form a team of 4 engineers, but the team must include at least one mechanical and at least one civil engineer. Tasks: Help the manager to: Determine the optimal concrete composition that maximizes durability within budget constraints and analyze the delivery schedule adequacy, providing recommendations with justification and judgement on resource allocation efficiency. Plan the site layout and optimal team composition, providing recommendations with justification on how these ensure structural integrity and project success.

multipleQuestions newQuestion What is the optimal concrete composition that maximizes durability within budget constraints? newAnswer To maximize durability, set the budget constraint \( 45,000x ≤ 450,000 \), leading to \( x ≤ 10 \) kg. The durability function \( D(x) = 28 + 2x - 0.1x^2 \) is maximized at \( x = 10 \) kg, yielding \( D(10) = 28 + 20 - 10 = 38 \) years. newQuestion How adequate is the delivery schedule of the additive? newAnswer The delivery schedule forms a geometric progression with a common ratio of 1.5, resulting in total delivery of 50 + 75 + 112.5 + 168.75 + 253.125 = 659.375 kg over five weeks, which exceeds the maximum requirement of 10 kg per cubic metre. newQuestion What is the optimal team composition of engineers? newAnswer The optimal team composition can be achieved by selecting 2 civil engineers and 2 mechanical engineers, or 3 civil engineers and 1 mechanical engineer, ensuring at least one of each type while maximizing expertise. newQuestion How does the site layout contribute to structural integrity and project success? newAnswer A well-planned site layout ensures efficient material flow, minimizes risks during construction, and facilitates easy access to anchoring points A and B, which is crucial for maintaining structural integrity and ensuring project success.