Final answer:
To maximize revenue from the sale of fishing permits, the country park should charge $84 per permit. This is calculated by setting up a revenue function based on the price increase and the decrease in the number of permits sold, then finding the vertex of the parabola that this function represents.
Step-by-step explanation:
To determine the price at which the country park should charge for a fishing permit to maximize revenue, we set up a revenue function. Let x be the number of $3 price increases. If they sell 300 permits at $75 each without any price increase, the original revenue is 300 times $75. With each $3 increase in price, they lose 10 customers. Therefore:
Revenue = (Number of Permits Sold) × (Price per Permit)
Revenue = (300 - 10x) × (75 + 3x)
To find the maximum revenue, we need to determine the vertex of the parabolic function defined by this revenue equation. The x-coordinate of the vertex can be found by -b/(2a) from the standard quadratic form ax^2 + bx + c. First, let's multiply out our revenue equation and get it in standard form:
Revenue = 22500 + 210x - 30x^2
Comparing this to the standard form, we identify a = -30, b = 210, and c = 22500. The x-coordinate of the vertex is -b/(2a):
x = -210 / (2 × -30)
x = -210 / -60
x = 3.5
Since x represents the number of $3 increases, and we can't have a half of a price increase, we round down to 3 (ensuring we don't surpass the maximum). Therefore, the park should increase the price by 3 × $3 = $9.
The maximum revenue price per permit is then $75 + $9, which is $84.