Approvimate the change in the volume of a right circular cylinder of foxed radus \( \mathrm{r}=19 \mathrm{~cm} \) when its height decreases from \( \mathrm{h}=11 \mathrm{~cm} \) to \( \mathrm{h}=10.9 \mathrm{~cm}\left(\mathrm{~V}(\mathrm{~h})=\mathrm{ur}^{2} \mathrm{~h}\right) \). Use a inear approxination.

When you want to approximate the change in the volume of a right circular cylinder using linear approximation, you can utilize the concept of calculus. The volume \( V \) of a cylinder is given by the formula \( V = \pi r^2 h \). If we know the height decreases from \( h = 11 \, \text{cm} \) to \( h = 10.9 \, \text{cm} \), we can define a small change in height \( \Delta h = -0.1 \, \text{cm} \). First, we find the derivative of the volume with respect to height: \[ \frac{dV}{dh} = \pi r^2 \] Substituting \( r = 19 \, \text{cm} \) gives: \[ \frac{dV}{dh} = \pi (19^2) = \pi (361) \approx 1134.11 \, \text{cm}^2 \] Now, to find the approximate change in volume \( \Delta V \): \[ \Delta V \approx \frac{dV}{dh} \cdot \Delta h = 1134.11 \cdot (-0.1) \approx -113.41 \, \text{cm}^3 \] So, the volume of the cylinder decreases by approximately \( 113.41 \, \text{cm}^3 \) as the height decreases from \( 11 \, \text{cm} \) to \( 10.9 \, \text{cm} \).