a diagonal of a cube measures $\sqrt{750}$ cm. the diagonal of a face measures $\sqrt{500}$ cm.

in centimeters, what is the length of an edge of the cube? round the answer to the nearest tenth.

centimeters

Question

a diagonal of a cube measures $\sqrt{750}$ cm. the diagonal of a face measures $\sqrt{500}$ cm.

in centimeters, what is the length of an edge of the cube? round the answer to the nearest tenth.

centimeters

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Pythagorean Theorem", "Cube Diagonals" ], "new_concepts": [], "current_concepts": [ "Pythagorean Theorem", "Cube Diagonals", "Square Root Estimation" ] } ## Relate the edge length to the diagonals using the Pythagorean theorem $$ \begin{aligned} d_{\text{face}}^2 &= s^2 + s^2 = 2s^2 \ d_{\text{cube}}^2 &= s^2 + d_{\text{face}}^2 = 3s^2 \end{aligned} $$ ## Solve for the edge length $s$ $$ \begin{aligned} d_{\text{face}} &= \sqrt{500} \implies d_{\text{face}}^2 = 500 \ 2s^2 &= 500 \ s^2 &= 250 \ s &= \sqrt{250} \end{aligned} $$ ## Calculate and round the decimal value $$ \begin{aligned} s &= \sqrt{250} \approx 15.811388 \ s &\approx 15.8 \text{ cm} \end{aligned} $$ A diagonal of a cube measures $\sqrt{750}$ cm. The diagonal of a face measures $\sqrt{500}$ cm. In centimeters, what is the length of an edge of the cube? Round the answer to the nearest tenth. 15.8 centimeters { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Cube Diagonals" ] }

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QUESTION IMAGE

Question

Explanation:

Relate the edge length to the diagonals using the Pythagorean theorem

Solve for the edge length \(s\)

Calculate and round the decimal value

Answer: