2024 ASRJC H1 Q5

A company produces three types of electronic products: Headphones, Gaming Controllers and Bluetooth Speakers.

• The total manufacturing cost of \(5\) Headphones, \(6\) Gaming Controllers and \(10\) Bluetooth Speakers is \(\$1400\).
• The total manufacturing cost of a Headphone and a Gaming Controller is twice the manufacturing cost price of a Bluetooth Speaker.
• The total manufacturing cost of \(10\) Gaming Controllers is \(\$131.25\) more than the total cost of \(5\) Headphones and \(5\) Bluetooth Speakers.

1. By writing down three linear equations, find the manufacturing cost of one Headphone.[4]

An economist advises the company on how to increase their profit.\[\text{Profit} = \text{selling price} - \text{manufacturing cost}\]

In a simple model, the economist suggested that the electronic products should be priced at \(\$80\) per electronic product.

2. Find the profit from the sale of \(70\) of each of Headphones, Gaming Controllers and Bluetooth Speakers.[2]

The company is trialing a new type of product, Wireless Earbuds. The economist predicts that the profit \(\$P\) will be related to the manufacturing cost (\(\$x\)) by the equation

\[P = 6\sqrt{x} - 0.8x\]

3. Sketch the graph of \(P\) against \(x\), stating the coordinates of the points of intersection with the \(x\)-axis.[2]
4. Use your calculator to estimate the maximum value of \(P\). State also the value of \(x\) for which this maximum value occurs.[2]
5. Given that the manufacturing cost of a pair of Wireless Earbuds is \(\$40\), find the selling price.[2]