question 38 ### Transcription of Educational Content**Problem 36:**A faucet is filling a hemispherical basin of diameter 60 cm with water at a rate of 2 L/min. Find the rate at which the water is rising in the basin when it is half full. [Use the following facts: 1 L is 1000 cm³. The volume of the portion of a sphere with radius $ r $ from the bottom to a height $ h $ is $ V = \pi (rh^2 - \frac{1}{3}h^3) $, as we will show in Chapter 5.]**Problem 37:**Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure $ P $ and volume $ V $ satisfy the equation $ PV = C $, where $ C $ is a constant. Suppose that at a certain instant the volume is 600 cm³, the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPa/min. At what rate is the volume decreasing at this instant?**Problem 38:**When air expands adiabatically (without gaining or losing heat), its pressure $ P $ and volume $ V $ are related by the equation $ PV^{1.4} = C $, where $ C $ is a constant. Suppose that at a certain instant the volume is 400 cm³ and the pressure is 80 kPa and is decreasing at a rate of 10 kPa/min. At what rate is the volume increasing at this instant?**Problem 39:**If two resistors with resistances $ R_1 $ and $ R_2 $ are connected in parallel, as in the figure, then the total resistance $ R $, measured in ohms ($ \Omega $), is given by\[\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\]If $ R_1 $ and $ R_2 $ are increasing at rates of 0.3 $ \Omega/s $ and 0.2 $ \Omega/s $, respectively, how fast is $ R $ changing when $ R_1 = 80 \, \Omega $ and $ R_2 = 100 \, \Omega $?**Diagram

To determine the rate at which volume is increasing

38. When air expands adiabatically (without gaining or losing zi heat), its pressure P and volume V are related by the equa- tion PV1.4 C, where C is a constant. Suppose that at a certain instant the volume is 400 cm³ and the pressure is 80 kPa and is decreasing at a rate of 10 kPa/min. At what hear rate is the volume increasing at this instant? =