The radius \( r \) of a circle is increasing at a rate of 2 centimeters per minute. Find the rate of change of the area (in \( \left.\mathrm{cm}{ }^{2} / \mathrm{min}\right) \) when \( r=36 \) centimeters. \( \square \) \[ \mathrm{cm}^{2} / \mathrm{min} \] Need Help? Ruad it

To find the rate of change of the area of the circle as the radius increases, we can use the formula for the area \( A \) of a circle, which is given by \( A = \pi r^2 \). To find the rate of change of the area \( \frac{dA}{dt} \), we can use the chain rule: \[ \frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt} \] First, we calculate \( \frac{dA}{dr} \): \[ \frac{dA}{dr} = 2\pi r \] Given \( r = 36 \) cm and \( \frac{dr}{dt} = 2 \) cm/min, we plug in these values: \[ \frac{dA}{dt} = 2\pi (36) \cdot 2 = 144\pi \, \text{cm}^2/\text{min} \] Now, if you were looking for an exciting area of exploration, back in Ancient Greece, mathematicians like Archimedes were studying the properties of circles, not just for mathematics but also for architecture and art! Circles represented perfection and eternity, making them a favorite among builders and artists alike. Additionally, understanding the geometric principles behind circles can lead to various real-world applications, such as designing roundabouts for efficient traffic flow, optimizing storage space in warehouses, or even creating artistic installations. Knowing how the area changes as the radius increases can also aid engineers in designing objects that require precise curvature, such as lenses or tanks!