a rectangle is constructed under the graph of $f(x) = e^{6x}$ with one corner at $(7,0)$ and one corner on the graph of $f(x) = e^{6x}$ where $0 \le x \le 7$.

find the exact value of $x$ that will maximize the area of the rectangle.

$x =$

Question

a rectangle is constructed under the graph of $f(x) = e^{6x}$ with one corner at $(7,0)$ and one corner on the graph of $f(x) = e^{6x}$ where $0 \le x \le 7$.

find the exact value of $x$ that will maximize the area of the rectangle.

$x =$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Optimization Problems", "Domain of Optimization", "Critical Points" ], "new_concepts": [], "current_concepts": [ "Optimization Problems", "Domain of Optimization", "Critical Points" ] } ## Formulate the area function Using the *Optimization Problems* and *Domain of Optimization* knowledge points $$ \begin{aligned} &\text{Width of the rectangle: } w = 7 - x\ &\text{Height of the rectangle: } h = f(x) = e^{6x}\ &A(x) = (7 - x)e^{6x},\quad x \in [0, 7] \end{aligned} $$ ## Find the derivative of the area function Using the *Optimization Problems* knowledge point $$ \begin{aligned} &A'(x) = \frac{d}{dx}[7 - x] \cdot e^{6x} + (7 - x) \cdot \frac{d}{dx}[e^{6x}]\ &A'(x) = -1 \cdot e^{6x} + (7 - x) \cdot 6e^{6x}\ &A'(x) = e^{6x}(-1 + 42 - 6x) = e^{6x}(41 - 6x) \end{aligned} $$ ## Determine the critical points Using the *Critical Points* and *Optimization Problems* knowledge points $$ \begin{aligned} &A'(x) = 0 \implies e^{6x}(41 - 6x) = 0\ &e^{6x} e 0 \implies 41 - 6x = 0 \implies x = \frac{41}{6} \end{aligned} $$ ## Verify the maximum value Using the *Optimization Problems* knowledge point $$ \begin{aligned} &\text{For } x < \frac{41}{6},\quad A'(x) > 0 \quad (\text{increasing})\ &\text{For } x > \frac{41}{6},\quad A'(x) < 0 \quad (\text{decreasing})\ &\text{Thus, } x = \frac{41}{6} \text{ maximizes } A(x). \end{aligned} $$ Find the exact value of $x$ that will maximize the area of the rectangle. $x =$ $\frac{41}{6}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Optimization Problems" ] }

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QUESTION IMAGE

Question

Explanation:

Formulate the area function

Find the derivative of the area function

Determine the critical points

Verify the maximum value

Answer: