the volume of a cube decreases at a rate of $8\text{ m}^3/\text{s}$. find the rate at which the side of the cube changes when the side of the cube is $2\text{ m}$. answer exactly or round to 2 decimal places.

the side of the cube changes at a rate of $\text{m}/\text{s}$

Question

the volume of a cube decreases at a rate of \$8\text{ m}^3/\text{s}\$. find the rate at which the side of the cube changes when the side of the cube is \$2\text{ m}\$. answer exactly or round to 2 decimal places.

the side of the cube changes at a rate of \$\text{m}/\text{s}\$

the side of the cube changes at a rate of   $\text{m}/\text{s}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Related Rates" ], "new_concepts": [], "current_concepts": [ "Related Rates", "Implicit Differentiation" ] } ## Relate volume and side length $$ V = s^3 $$ ## Differentiate with respect to time $$ \frac{dV}{dt} = 3s^2 \frac{ds}{dt} $$ ## Substitute given values and solve $$ \begin{aligned} -8 &= 3(2)^2 \frac{ds}{dt} \ -8 &= 12 \frac{ds}{dt} \ \frac{ds}{dt} &= -\frac{2}{3} \approx -0.67 \end{aligned} $$ The volume of a cube decreases at a rate of $8\text{ m}^3/\text{s}$. Find the rate at which the side of the cube changes when the side of the cube is $2\text{ m}$. Answer exactly or round to 2 decimal places. The side of the cube changes at a rate of $-2/3$ $\text{m}/\text{s}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Related Rates" ] }

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Question

Explanation:

Relate volume and side length

Differentiate with respect to time

Substitute given values and solve

Answer: