select the correct answer.

naomi is building a circuit board. the final microchip should have a surface area of 864 square millimeters. the height of the microchip can be a maximum of 4 millimeters. what are the maximum dimensions of the microchip she can use?

- 12 mm and 24 mm
- 9 mm and 18 mm
- 8 mm and 16 mm
- 6 mm and 12 mm

Question

select the correct answer.

naomi is building a circuit board. the final microchip should have a surface area of 864 square millimeters. the height of the microchip can be a maximum of 4 millimeters. what are the maximum dimensions of the microchip she can use?

- 12 mm and 24 mm
- 9 mm and 18 mm
- 8 mm and 16 mm
- 6 mm and 12 mm

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Surface Area of Prisms" ], "new_concepts": [ "Surface Area of Rectangular Prisms" ], "current_concepts": [ "Surface Area of Prisms", "Surface Area of Rectangular Prisms" ] } ## Set up the surface area equation for the rectangular prism $$ \begin{aligned} &\text{Length } l = 2x \ &\text{Width } w = x \ &\text{Height } h = 4 \ &S = 2(lw + lh + wh) = 864 \end{aligned} $$ ## Solve the quadratic equation for x $$ \begin{aligned} &2(2x^2 + 8x + 4x) = 864 \ &2x^2 + 12x - 432 = 0 \ &x^2 + 6x - 216 = 0 \ &(x + 18)(x - 12) = 0 \ &x = 12 \quad (x > 0) \end{aligned} $$ ## Calculate the maximum dimensions $$ \begin{aligned} &w = x = 12\text{ mm} \ &l = 2x = 24\text{ mm} \end{aligned} $$

(A) 12 mm and 24 mm

(B) 9 mm and 18 mm

(C) 8 mm and 16 mm

(D) 6 mm and 12 mm

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Surface Area of Rectangular Prisms" ] }

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QUESTION IMAGE

Question

Explanation:

Set up the surface area equation for the rectangular prism

Solve the quadratic equation for x

Calculate the maximum dimensions

Answer: