The annual yield per lemon tree is fairly constant at 320...

The annual yield per lemon tree is fairly constant at 320 pounds when the number of trees per acre is 50 or fewer. For each additional tree over $50,$ the annual yield per tree for all trees on the acre decreases by 4 pounds due to overcrowding. Find the number of trees that should be planted on an acre to produce the maximum yield. How many pounds is the maximum yield?

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Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form ax² + bx + c that produce parabolic graphs. They are central to optimization problems because their shape guarantees a single maximum or minimum value, depending on the coefficient of the squared term.

Vertex of a Parabola

The vertex of a parabola represents the point where a quadratic function reaches its maximum or minimum value. It can be calculated using the formula -b/(2a), which is an essential tool in determining the optimal values in various problems, such as maximizing yield or minimizing cost.

Optimization with Trade-Offs

Many real-world scenarios involve a trade-off where improving one variable results in the degradation of another. This balance is often modeled with quadratic functions to find an optimal point that maximizes or minimizes the desired outcome, making tools like vertex calculation crucial in decision-making.

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