Diagonals If x, y, and z are lengths of the e...

Question

Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).

b. How is ds/dt related to dy/dt and dz/dt if x is constant?

Accepted Answer

Welcome back, everyone. A cylindrical tank with radius R and height H is being filled with water. If the radius is constant, how is the rate of change of the volume related to the rate of change of the height? For this problem, let's recall that the volume of a cylinder V is equal to pi R2 H, where R is radius and H is height, which we know about. The parameters from this equation according to the context of the problem. Well, it says that. The radius is constant, right? So R is a constant. And each is variable. So what we're going to do is simply understand that B is a function of H, but H is a function of time, right? So we have a composite function B of H of T. And because we have a composite function, if we want to differentiate it, and the reason why we want to differentiate it is because we're looking for the rate of change of volume, right? So essentially we're looking for DV divided by DT, the rate of change of volume with respect to time, and essentially because we have a composite function, we have to apply the chain rule. First of all, we are differentiating volume. With respect to height, so DV divided by DH, and according to the chain rule, we have to multiply by the rate of change of height with respect to time. So DH divided by DT. Now, let's look at the formula. And let's identify DV divided by DH. So what we want to do is simply differentiate pi R2 H. We know that i multiplied by R2 is a constant, so we simply need to identify the derivative of H with respect to H, which is one, right? DH divided by DH gives us 1, so we get DV divided by DH is equal to pi R2. And now we're looking for the relationship between the rate of change of the height, DH divided by DT and the rate of change of volume DV divided by DT. So we have identified the relationship using the constant multiple. We can say that DV divided by DT. Is equal to the product between pi squared our constant multiple. And DH divided by DT. And this equation tells us the relationship between the two parameters. Thank you for watching.

Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).
b. How is ds/dt related to dy/dt and dz/dt if x is constant?

Key Concepts

Partial Derivatives

Chain Rule

Pythagorean Theorem in 3D

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Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).b. How is ds/dt related to dy/dt and dz/dt if x is constant?