a farmer finds that if she plants 60 trees per acre, each tree will yield 70 bushels of fruit. she estimates that for each additional tree planted per acre, the yield of each tree will decrease by 4 bushels. how many trees should she plant per acre to maximize her harvest?

trees

give your answer rounded to the nearest whole number.

Question

a farmer finds that if she plants 60 trees per acre, each tree will yield 70 bushels of fruit. she estimates that for each additional tree planted per acre, the yield of each tree will decrease by 4 bushels. how many trees should she plant per acre to maximize her harvest?

trees

give your answer rounded to the nearest whole number.

trees

give your answer rounded to the nearest whole number.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Optimization Problems", "Quadratic Optimization" ], "new_concepts": [], "current_concepts": [ "Optimization Problems", "Quadratic Optimization" ] } ## Formulate the yield function $$ \begin{aligned} x &= \text{number of additional trees per acre}\ T(x) &= 60 + x \quad (\text{total trees per acre})\ Y(x) &= 70 - 4x \quad (\text{yield per tree in bushels})\ H(x) &= T(x) \cdot Y(x) = (60 + x)(70 - 4x) \quad (\text{total harvest per acre})\ H(x) &= 4200 - 240x + 70x - 4x^2 = -4x^2 - 170x + 4200 \end{aligned} $$ ## Find the critical point $$ \begin{aligned} H'(x) &= -8x - 170 = 0\ x &= -\frac{170}{8} = -21.25 \end{aligned} $$ ## Calculate the optimal number of trees $$ \begin{aligned} T_{\text{optimal}} &= 60 + x = 60 - 21.25 = 38.75\ T_{\text{rounded}} &\approx 39 \end{aligned} $$ A farmer finds that if she plants 60 trees per acre, each tree will yield 70 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 4 bushels. How many trees should she plant per acre to maximize her harvest? 39 trees { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Optimization Problems" ] }

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QUESTION IMAGE

Question

Explanation:

Formulate the yield function

Find the critical point

Calculate the optimal number of trees

Answer: