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).

c. How are dx/dt, dy/dt, and dz/dt related if s is constant?

Accepted Answer

Welcome back, everyone. A cones volume V changes over time where V equals 1/3 by R2 H, and both are radius and H height are functions of time. How are the divided by DT and D H divided by DT related if V is constant. So let's begin with the equation V equals 1/3 by R2 H. We know that R. It is a function of time and we also know that H is a function of time. So we have a composite function, right? And this means that if we want to differentiate DV divided by the T, we're going to apply the chain rule. First of all, let's factor out the constant 1/3 pi. And this means that we are differentiating the product R2 multiplied by H. So let's apply the product rule. We're going to get 13 pi multiplied by the derivative of R2, which is 2R, but R is a function of time, so we're also multiplying by the r divided by D T, and this is multiplied by our second function H. And then we're adding our first function are squared multiplied by the derivative of h. With respect to age. Which is simply one right multiplied by the derivative of H with respect to time because H is a function of time. And now what can we tell? Well, we can sell the DV divided by DT is 0 because it's a constant. And this gives us 0 equals. Now, if we distribute, we're going to get 2/3 by R D R divided by DT multiplied by H. Plus 1/3. I R squared, D H divided by DT and from here we can identify our final relationship. First of all, we can multiply both sides by 3, which eliminates the denominator. And we can divide both sides by pi, which eliminates pi completely, right? We can also divide both sides by R because R must be positive. It's non-zero, right? And also positive, so we can divide both sides by R and rearrange the equation. Which tells us that negative to HDR divided by DT is equal to. R D H divided by D T And that we can solve for the R divided by DT by dividing both sides by negative2 H, which gives us DR divided by the T equals negative R divided by 2 H. DH divided by DT and this is our final relationship between the two derivatives. 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).
c. How are dx/dt, dy/dt, and dz/dt related if s is constant?

Key Concepts

Partial Derivatives

Chain Rule

Implicit Differentiation

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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).c. How are dx/dt, dy/dt, and dz/dt related if s is constant?