The surface area of a cube is given by $S=6 x^2$ where $x$ is the length of its edges measured in mm. The surface area is decreasing at a constant rate of 2 $\text{mm}^2/\text{second}$. At the instant when $x=8$, at what rate (in $\text{mm}/\text{second}$) is $x$ changing?

Question

04-11-2026
Mathematics

Answered

The surface area of a cube is given by $S=6 x^2$ where $x$ is the length of its edges measured in mm. The surface area is decreasing at a constant rate of 2 $\text{mm}^2/\text{second}$. At the instant when $x=8$, at what rate (in $\text{mm}/\text{second}$) is $x$ changing?

Verified Answer :

Related Questions

The number of milligrams of a certain medicine a veterinarian gives to a dog varies directly with the weight of the dog If the veterinarian gives a 30pound dog

A shelf has 20 bags of potatoes Two of the bags have potatoes starting to go bad What is the expected probability of the complement of the event bad bags of pot

the controversy over Missouris request for statehood A Did not affect the balance of power in Congress B Was settled by the Missouri comprise C Precipitated the

If Jimin is 59 and Namjoon is 511 what is the difference in height You dont have to answer this

DUE AN HOUR AGO Which of the following sets of numbers is a Pythagorean triple 14 40 50 7 24 31 7 20 27 14 48 50

how do scientists analyze empirical evidence

how do you simplify the ratio 287

Which best explains how the structure of the office of the president helps fulfill the offices role A The office is led by the chief of staff who serves as a ke

john is twice as old as mary the sum of their ages is 21 how old is mary let j johns age and m marys age select the system equations that represents the problem

What led to Iraq being ruled by a series of military dictators A The 14 July Revolution B The Arab Revolt C The Gulf War D The IranIraq War

GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T23:10:52-05:00

Use the surface area formula $S = 6x^2$ to relate the change in surface area to the change in edge length.
Differentiate with respect to time using the chain rule to obtain $\frac{dS}{dt} = 12x \frac{dx}{dt}$.
Substitute the given rate $\frac{dS}{dt} = -2 \, \text{mm}^2/\text{second}$ and the edge length $x = 8 \, \text{mm}$ into the derivative equation.
Solve the resulting equation to find that the rate of change is $\boxed{-\frac{1}{48} \, \text{mm}/\text{second}}$.

Explanation

Problem Analysis

The surface area $S$ of a cube with edge length $x$ is given by the formula $S = 6x^2$. We are given that the surface area $S$ is decreasing at a constant rate of $2 \, \text{mm}^2/\text{second}$, which means the rate of change of $S$ with respect to time $t$ is $\frac{dS}{dt} = -2 \, \text{mm}^2/\text{second}$. We want to find the rate at which the edge length $x$ is changing, $\frac{dx}{dt}$, at the specific instant when the edge length $x = 8 \, \text{mm}$.

Differentiating the Surface Area Formula

To find the rate of change of the edge length, we relate the surface area $S$ to the edge length $x$ through the given equation $S = 6x^2$. Since both $S$ and $x$ are functions of time $t$, we differentiate both sides of this equation with respect to $t$ using the chain rule:
\[\frac{dS}{dt} = \frac{d}{dt}(6x^2) = 12x \cdot \frac{dx}{dt}\]

Substituting Values and Solving for the Rate of Change

With the derivative established as $\frac{dS}{dt} = 12x \cdot \frac{dx}{dt}$, we can isolate $\frac{dx}{dt}$ to solve for the rate of change of the edge length:
\[\frac{dx}{dt} = \frac{\frac{dS}{dt}}{12x}\]
Now, we substitute the known values into this equation: $\frac{dS}{dt} = -2 \, \text{mm}^2/\text{second}$ and $x = 8 \, \text{mm}$.
\[\frac{dx}{dt} = \frac{-2}{12 \cdot 8} = \frac{-2}{96}\]

Final Result and Interpretation

Simplifying the fraction $\frac{-2}{96}$, we get $\frac{-1}{48}$. Therefore, the rate at which the edge length $x$ is changing at the instant $x = 8 \, \text{mm}$ is $-\frac{1}{48} \, \text{mm}/\text{second}$. The negative sign indicates that the edge length is decreasing. Thus, the rate is $\boxed{-\frac{1}{48} \, \text{mm}/\text{second}}$.

Examples

Understanding the rate of change is essential in engineering and manufacturing. For instance, if a thermal expansion process causes a cube-shaped component to shrink uniformly as it cools, an engineer needs to predict how quickly each side length will shorten. By knowing the total surface area's rate of decrease, they can use calculus to determine the exact speed at which the dimensions are changing, ensuring that the part remains within specified tolerances during the cooling cycle.