you are planning to make an open rectangular box from a 17-in.-by-33-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. what are the dimensions of the box of largest volume you can make this way, and what is its volume?

the dimensions of box of maximum volume are
(round to the nearest hundredth as needed. use a comma to separate answers as needed.)

Question

you are planning to make an open rectangular box from a 17-in.-by-33-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. what are the dimensions of the box of largest volume you can make this way, and what is its volume?

the dimensions of box of maximum volume are
(round to the nearest hundredth as needed. use a comma to separate answers as needed.)

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", "Applied Derivatives" ], "new_concepts": [], "current_concepts": [ "Optimization Problems", "Applied Derivatives" ] } ## Formulate the volume function $$ \begin{aligned} &w = 17 - 2x,\quad l = 33 - 2x,\quad h = x\ &V(x) = x(17 - 2x)(33 - 2x) = 4x^3 - 100x^2 + 561x\ &\text{Domain: } 0 < x < 8.5 \end{aligned} $$ ## Find the critical points $$ \begin{aligned} &V'(x) = 12x^2 - 200x + 561 = 0\ &x = \frac{200 \pm \sqrt{(-200)^2 - 4(12)(561)}}{24}\ &x = \frac{200 \pm \sqrt{40000 - 26928}}{24} = \frac{200 \pm \sqrt{13072}}{24}\ &x \approx \frac{200 \pm 114.3328}{24}\ &x_1 \approx 13.10 \quad (\text{outside domain})\ &x_2 \approx 3.57 \quad (\text{inside domain}) \end{aligned} $$ ## Calculate the optimal dimensions $$ \begin{aligned} &h = x \approx 3.57\text{ in.}\ &w = 17 - 2(3.5695) \approx 9.86\text{ in.}\ &l = 33 - 2(3.5695) \approx 25.86\text{ in.} \end{aligned} $$ The dimensions of box of maximum volume are 3.57, 9.86, 25.86 in. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Optimization Problems" ] }

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QUESTION IMAGE

Question

Explanation:

Formulate the volume function

Find the critical points

Calculate the optimal dimensions

Answer: