Final answer:
Matthew should use calculus to find the price that maximizes sales by setting the derivative of the revenue function to zero to identify the critical points.
Assuming no change in costs, this will give him the price that maximizes revenue, which might also contribute to maximizing profits if marginal costs are considered.
Step-by-step explanation:
To determine the price that maximizes sales for the surfboard company's t-shirts, Matthew needs to find the maximum point of the revenue function given by r = -3p2 + 60p + 1060, where p represents the price of the company's product.
This can be achieved using calculus to find the derivative of the revenue function and setting it equal to zero to find the critical points.
Matthew should calculate the derivative of the revenue function with respect to p, which is dr/dp = -6p + 60. Setting this equal to zero gives the price p where marginal revenue equals zero, which is typically the point where total revenue is maximized. Solving -6p + 60 = 0 gives p = 10.
This would be the price to maximize revenue, assuming costs do not change with quantity sold.
The principle of maximizing profits can also be relevant, where firms aim to produce where marginal revenue equals marginal cost (MR=MC).
Working through the numbers can help understand this principle and ensure that Matthew does not only increase revenue but also maximizes profits.
It's important to remember that this analysis assumes the absence of external factors that could affect demand or the sales of t-shirts at different price points. Such factors might include customer preferences, market competition, and production costs.