Exercise · ZW-RELATED-RATES-007 · ★★☆ Intermediate

A rectangle has length \(x=4\) m and width \(y=3\) m. The length grows at \(1\) m/s and the width shrinks at \(2\) m/s. How fast is the area changing at that instant? (in m²/s)

Choose one option.

Show answer
  • C. \(-5\)

Explanation

With \(A=xy\) we get \(\dfrac{dA}{dt}=y\,\dfrac{dx}{dt}+x\,\dfrac{dy}{dt}=3(1)+4(-2)=-5\) m²/s, so the area decreases.

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Review the concept: Related rates

In a related rates problem several quantities change with time \(t\) and are linked by an equation. The equation is differentiated with respect to \(t\) (chain rule) and the known values are substituted.

\[A=\pi r^{2}\ \Rightarrow\ \frac{dA}{dt}=2\pi r\,\frac{dr}{dt}\]

More practice problems: related rates

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