The volume of a cube decreases at a rate of 0.5 ft^3 / min . What is the rate of change of the s

The volume of a cube decreases at a rate of 0.5 ft^3 / min ....

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Volume of a Cube

The volume of a cube is determined by the formula V = s³, where 's' is the side length of the cube. This fundamental geometric relationship provides the basis for connecting changes in the cube's side length to its volume, an essential step in solving problems involving volume changes over time.

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time when another related quantity changes with time. This approach is common in calculus, and it requires establishing a connection between the two variables, differentiating with respect to time, and then solving for the unknown rate.

Chain Rule

The chain rule is a technique in differentiation used to compute the derivative of a composite function. In the context of a cube's volume and side length, it allows us to differentiate V = s³ with respect to time by linking the rate of change of volume (dV/dt) to the rate of change of the side length (ds/dt) through the derivative dV/ds.

Transcript

00:01 Hi everyone.

00:02 For this problem, we're asked to find the rate of change or the sides of a cube, and we are given that the volume of this cube is decreasing at 0 .5 cubic feet per minute.

00:15 So let's write that out.

00:17 The volume, dvd, the rate of change of the volume, and i'm going to put it into negative one -half here.

00:27 That negative comes from decreasing.

00:29 And this is cubic.

00:32 Feet per minute.

00:36 All right, we are also given that the sides equal 12 feet, and they would like to know the rate of change of the sides here, or dx, dt.

00:53 That's their, that's what they're looking for.

00:59 All right.

00:59 So let's see if we can figure that out.

01:02 The volume of the cube is given by length and width times height.

01:06 In this case, they are all the same.

01:08 There x x times x times x is equal to x cubed so now let's differentiate both sides of this equation with respect to time on the left i will get bdd t one of our knowns and that is equal to we get some chain roll here bring a three down and it's an x squared then i've got a d x d t which is what we're looking for all right so let's put in our stuff here.

01:46 So dvdt is negative, 1 half, and that is in cubic feet per minute.

02:00 All right, and that is equal to three times 144...

Please give Ace some feedback