Let l be the length of a diagonal of a rectangle whose sides...

Let $l$ be the length of a diagonal of a rectangle whose sides have lengths $x$ and $y,$ and assume that $x$ and $y$ vary with time. (a) How are $d l / d t, d x / d t,$ and $d y / d t$ related? (b) If $x$ increases at a constant rate of $\frac{1}{2} \mathrm{ft} / \mathrm{s}$ and $y$ decreases at a constant rate of $\frac{1}{4} \mathrm{ft} / \mathrm{s}$, how fast is the size of the diagonal changing when $x=3 \mathrm{ft}$ and $y=4 \mathrm{ft} ?$ Is the diagonal increasing or decreasing at that instant?

Let $l$ be the length of a diagonal of a rectangle whose
sides have lengths $x$ and $y,$ and assume that $x$ and $y$ vary
with time.
(a) How are $d l / d t, d x / d t,$ and $d y / d t$ related?
(b) If $x$ increases at a constant rate of $\frac{1}{2} \mathrm{ft} / \mathrm{s}$ and $y$ decreases at a constant rate of $\frac{1}{4} \mathrm{ft} / \mathrm{s}$, how fast is the size of the diagonal changing when $x=3 \mathrm{ft}$ and
$y=4 \mathrm{ft} ?$ Is the diagonal increasing or decreasing at that instant?

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Key Concepts

Pythagorean Theorem

In a rectangle, the diagonal forms the hypotenuse of a right triangle whose legs are the side lengths. The Pythagorean theorem, which states that the sum of the squares of the legs equals the square of the hypotenuse, provides the core relationship between the rectangle's sides and its diagonal.

Related Rates

Related rates problems involve determining the rates at which quantities change with respect to time when these quantities are linked by an equation. In such problems, knowing the rate of change of one variable helps in finding the rate of change of another by differentiating the relation between them.

Implicit Differentiation

Implicit differentiation is a method used when variables are interrelated by an equation. It involves differentiating both sides of an equation with respect to time, even if one variable is not explicitly solved in terms of the other, to relate their rates of change.

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. When a variable is indirectly dependent on time through other variables, the chain rule enables the correct computation of its derivative with respect to time.

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