Question 250 part 5 (Section B style Question) – Kaplan exam kit

Hi John, I hope you are well.

I was wondering if you could help me with a question, I’m having trouble understanding how to work out this type of question. The question is –

What is the optimum selling price at the profit maximising level of sales?

They provide the following information in the question –
Market research has revealed that the maximum demand for Jewel Co’s earphones in the USA is 72,000 units per year, and that demand will reduce by 8000 units for every $5 that the selling price is increased. Jewel Co has calculated the profit maximising level of sales for its earphones for the coming year is 32,000 units.

From my understanding we have to use the price demand equation to work this out. I was able to get as far as working out that b=5/8000 and that Q would be 32,000. I was unsure how to continue from here as I couldn’t seem to work out what ‘a’ would be. I understand the first 2 lines of kaplan’s solution below but then I’m confused as to how they worked out the remainder.

The solution in the kaplan exam kit is as follows:
P=0, demand (Q)=72,000 units
P=5, Q=72,000 – 8000 =64,000 units
Q=72,000-5P (P is the selling price in $ because demand will drop by 8000 units for every $5 increase in the selling price)
If the optimum quantity Q=32,000 units, P=5/8000 (72000-32000units) = $25.

Please could you kindly help me understand how to work this question out? Please let me know if any further information is needed as the detail to the question was quite lengthy but I have included what I thought would be relevant.

Thank you in advance 🙂
Bhavni