The sides of a square increase in length at a rate of \( 5 \mathrm{~m} / \mathrm{sec} \). a. At what rate is the area of the square changing when the sides are 19 m long? b. At what rate is the area of the square changing when the sides are 27 m long? a. Write an equation relating the area of a square, A, and the side length of the square, s. A \( =\mathrm{s}^{2} \) Differentiate both sides of the equation with respect to t . (

When differentiating \( A = s^2 \) with respect to \( t \), we use the chain rule: \( \frac{dA}{dt} = 2s \frac{ds}{dt} \). This means that if the side length \( s \) is changing, it will affect the area as well. Using \( \frac{ds}{dt} = 5 \mathrm{~m/sec} \), we can plug in our values to find how fast the area \( A \) is increasing. For part (a), when the side length is 19 m: \[ \frac{dA}{dt} = 2(19)(5) = 190 \text{ m}^2/\text{sec}. \] For part (b), when the side length is 27 m: \[ \frac{dA}{dt} = 2(27)(5) = 270 \text{ m}^2/\text{sec}. \] Now, wasn't it fascinating to see how changes in size lead to exponential changes in area? That's the power of geometry and calculus working hand-in-hand! If you were to visualize this, picture a small square slowly swelling into a larger one, and as the corners stretch further apart, the inner area fills like a balloon. Fun fact: This concept is not just confined to squares—it's crucial in various fields such as architecture and landscape design when ensuring ample space is allocated efficiently as structures grow!