At a certain vineyard it is found that each grape vine...

At a certain vineyard it is found that each grape vine produces about 10 pounds of grapes in a season when about 700 vines are planted per acre. For each additional vine that is planted, the production of each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by $$ A(n)=(700+n)(10-0.01 n) $$ where $n$ is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

At a certain vineyard it is found that each grape vine produces about 10 pounds of grapes in a season when about 700 vines are planted per acre. For each additional vine that is planted, the production of each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by
$$
A(n)=(700+n)(10-0.01 n)
$$
where $n$ is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form ax² + bx + c and are used to model relationships where two variables interact in a nonlinear manner. They often represent phenomena where increasing one variable initially increases the outcome until a certain point after which further increases decrease the outcome, making them ideal for representing production scenarios with trade-offs.

Vertex of a Parabola

The vertex of a parabola is the point where the function reaches its maximum or minimum value, depending on the direction in which the parabola opens. In optimization problems modeled by quadratic functions, finding the vertex using the formula -b/(2a) is crucial because it identifies the optimal condition, such as the maximum production point.

Optimization in Mathematical Modeling

Optimization in mathematical modeling involves determining the best possible outcome (like maximizing production or minimizing cost) by analyzing how variables interact within a given model. This process frequently employs quadratic functions and derivative-based methods to identify the point at which the desired outcome is optimized.

Transcript

Please give Ace some feedback