1. **Problem statement:** Given the formula for the speed of sound $$s = 389\sqrt{\frac{T}{m}}$$ where $s$ is speed of sound in km/h, $T$ is temperature in Kelvin, and $m$ is molecular mass in g/mol, solve the following: 2. **Formula and rules:** The formula relates speed, temperature, and molecular mass through a square root. To solve for $s$, substitute values directly. To solve for $T$, isolate $T$ by squaring both sides and rearranging. --- ### a. Speed of sound at altitude 4.9 km Given: $T=256$, $m=28.8$ Calculate: $$s = 389\sqrt{\frac{256}{28.8}}$$ Calculate the fraction inside the root: $$\frac{256}{28.8} \approx 8.8889$$ Calculate the square root: $$\sqrt{8.8889} \approx 2.9814$$ Multiply by 389: $$s = 389 \times 2.9814 \approx 1159.6$$ **Answer:** The speed of sound is approximately **1159.6 km/h**. --- ### b. Speed of sound in helium Given: $T=300$, $m=4.0$ Calculate: $$s = 389\sqrt{\frac{300}{4.0}}$$ Calculate the fraction inside the root: $$\frac{300}{4.0} = 75$$ Calculate the square root: $$\sqrt{75} \approx 8.6603$$ Multiply by 389: $$s = 389 \times 8.6603 \approx 3367.7$$ **Answer:** The speed of sound in helium is approximately **3367.7 km/h**. --- ### c. Rearrange formula to isolate $T$ and find temperature for $s=1000$ km/h Start with: $$s = 389\sqrt{\frac{T}{m}}$$ Divide both sides by 389: $$\frac{s}{389} = \sqrt{\frac{T}{m}}$$ Square both sides: $$\left(\frac{s}{389}\right)^2 = \frac{T}{m}$$ Multiply both sides by $m$: $$T = m \left(\frac{s}{389}\right)^2$$ Substitute $s=1000$, $m=28.8$: $$T = 28.8 \times \left(\frac{1000}{389}\right)^2$$ Calculate inside the parentheses: $$\frac{1000}{389} \approx 2.5707$$ Square it: $$2.5707^2 \approx 6.6095$$ Multiply by 28.8: $$T = 28.8 \times 6.6095 \approx 190.3$$ **Answer:** The temperature is approximately **190.3 K** when the speed of sound is 1000 km/h. --- **Summary:** - a) Speed of sound at 256 K and 28.8 g/mol is about 1159.6 km/h. - b) Speed of sound in helium at 300 K is about 3367.7 km/h. - c) Temperature for speed 1000 km/h and molar mass 28.8 g/mol is about 190.3 K.