the volume of a cube is decreasing at a rate of $320\text{ mm}^3/\text{s}$.

step 2 of 4: what is the rate of change of the cubes edge length $x$ when the edges are $80\text{ mm}$ long? write the exact answer. do not round.

answer

$\frac{dx}{dt} = $ $\text{mm}/\text{s}$

Question

the volume of a cube is decreasing at a rate of $320\text{ mm}^3/\text{s}$.

step 2 of 4: what is the rate of change of the cubes edge length $x$ when the edges are $80\text{ mm}$ long? write the exact answer. do not round.

answer

$\frac{dx}{dt} = $   $\text{mm}/\text{s}$

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": [ "Related Rates", "Volume of a Cube" ], "new_concepts": [], "current_concepts": [ "Related Rates", "Volume of a Cube", "Chain Rule Differentiation" ] } ## Identify given values and rates Using the *Related Rates* and *Volume of a Cube* knowledge points $$ \begin{aligned} &V = x^3\ &\frac{dV}{dt} = -320\text{ mm}^3/\text{s}\ &x = 80\text{ mm} \end{aligned} $$ ## Differentiate with respect to time Using the *Related Rates* and *Volume of a Cube* knowledge points $$ \frac{dV}{dt} = 3x^2 \frac{dx}{dt} $$ ## Solve for the rate of change of the edge length Using the *Related Rates* and *Volume of a Cube* knowledge points $$ \begin{aligned} -320 &= 3(80)^2 \frac{dx}{dt}\ -320 &= 3(6400) \frac{dx}{dt}\ -320 &= 19200 \frac{dx}{dt}\ \frac{dx}{dt} &= \frac{-320}{19200} = -\frac{1}{60} \end{aligned} $$ What is the rate of change of the cube's edge length $x$ when the edges are $80\text{ mm}$ long? Write the exact answer. Do not round. $\frac{dx}{dt} =$ $-\frac{1}{60}$ $\text{mm/s}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Related Rates" ] }

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QUESTION IMAGE

Question

Explanation:

Identify given values and rates

Differentiate with respect to time

Solve for the rate of change of the edge length

Answer: