Final answer:
To maximize income for Water World, the admission cost should be increased by $0.75, calculated by finding the vertex of the quadratic equation representing the income as a function of price increase and visitor numbers.
Step-by-step explanation:
To find the amount by which the admission cost to Water World should be increased to obtain maximum income, we can create a quadratic equation to represent the income based on the number of visitors and admission cost.
Step 1: Define variables and equation
Let x represent the increase in dollars to the current admission cost of $7.50. The new admission cost would be (7.50 + x) dollars.
With each $1 increase, the number of visitors decreases by 25. The new average number of visitors per day would be (225 - 25x) visitors.
Step 2: Formulate the income
Income, I, is equal to the admission cost multiplied by the number of visitors, which is I = (7.50 + x)(225 - 25x).
Step 3: Expand and maximize the quadratic equation
I = 1687.5 - 187.5x + 225x - 25x² = -25x² + 37.5x + 1687.5. This is a quadratic equation that opens downward, meaning it has a maximum point at its vertex.
The formula for the x-coordinate of the vertex of a parabola y = ax² + bx + c is -b/(2a). In this scenario, a = -25 and b = 37.5.
Step 4: Find the vertex
The x-coordinate of the vertex (maximum income) is -37.5 / (2 × -25) = 0.75. Therefore, the admission cost should be increased by $0.75 to maximize income.