Question

The sides of the rectangle above increase in such a way that (dz)/(dt) = 1 and (dx)/(dt) = 3(dy)/(dt). At the instant when x = 4 and y = 3, what is the value of (dx)/(dt)?

Answer 1

`\[ (dx)/(dt) = 3 \]`

In related rates, using the chain rule and given information, we find
`\((dx)/(dt)\)` as three times
`\((dy)/(dt)\)` with a final result of 3.

Step-by-step explanation:

In related rates problems, we often use the chain rule to express rates of change in terms of one another. Here, we are given
`\((dz)/(dt) = 1\)`and
`\((dx)/(dt) = 3(dy)/(dt)\)`. Using the chain rule,
`\((dz)/(dt) = (dz)/(dx) \cdot (dx)/(dt)\)`. Since
`\((dz)/(dt) = 1\), it implies \((dz)/(dx) = 1\)`.

Now, considering a rectangle where (x) and (z) are adjacent sides, we use
`\((dz)/(dx) = (dz)/(dy) \cdot (dy)/(dx)\). Given \((dz)/(dx) = 1\)`, it implies
`\((dy)/(dx) = 1\)`.

Finally, we're given
`\((dx)/(dt) = 3(dy)/(dt)\)`. Substituting
`\((dy)/(dx) = 1\),` we get
`\((dx)/(dt) = 3\)`.

Understanding related rates problems in calculus involves applying the chain rule to express the rates of change of variables with respect to time. This method helps establish relationships between different rates, allowing for the solution of complex problems in a systematic manner.

Answer 2

The value of
`\((dx)/(dt)\).` is 1 unit per time.

To find the value of
`\((dx)/(dt)\) given that \((dz)/(dt) = 1\) and \((dx)/(dt) = 3(dy)/(dt)\),` we need to understand the relationship between the variables
`\(x\), \(y\), and \(z\), where \(z\)`is likely the diagonal of the rectangle, and x and y are its sides.

Step 1: Relationship between
`\(x\), \(y\), and \(z\)`

For a rectangle, the diagonal z is related to the sides x and y by the Pythagorean theorem:

`\[ z^2 = x^2 + y^2 \]`

Step 2: Differentiate the Equation with Respect to Time t

Differentiating both sides of the equation with respect to t, we get:

`\[ 2z(dz)/(dt) = 2x(dx)/(dt) + 2y(dy)/(dt) \]`

Simplifying this, we get:

`\[ z(dz)/(dt) = x(dx)/(dt) + y(dy)/(dt) \]`

Step 3: Substitute the Given Rates and Solve for
`\((dx)/(dt)\)`

Given that
`\((dz)/(dt) = 1\) and \((dx)/(dt) = 3(dy)/(dt)\),`we substitute these into the equation along with the values of x = 4 and y = 3 at the instant we are interested in. Also, we can calculate z using the Pythagorean theorem for the given values of x and y.

Let's perform these calculations to find
`\((dx)/(dt)\).`

At the instant when
`\( x = 4 \) and \( y = 3 \), the value of \((dx)/(dt)\) is \( 1 \)`(unit per time). This means the side x of the rectangle is increasing at a rate of 1 unit per time unit.