1. **State the problem:** We are given that the speed $v$ of a vehicle is directly proportional to the power $p$ and the square of the radius $r$ of its wheels. Given $v=40$ m/s when $p=10$ kW and $r=0.4$ m, find $v$ when $p=16$ kW and $r=0.5$ m. 2. **Write the formula for direct proportionality:** $$v = k p r^2$$ where $k$ is the constant of proportionality. 3. **Find the constant $k$ using the initial conditions:** $$40 = k \times 10 \times (0.4)^2$$ Calculate $(0.4)^2$: $$0.4^2 = 0.16$$ So, $$40 = k \times 10 \times 0.16$$ $$40 = 1.6 k$$ Divide both sides by 1.6: $$k = \frac{40}{1.6}$$ Show cancellation: $$k = \frac{\cancel{40}}{\cancel{1.6}} \times \frac{1}{1} = 25$$ 4. **Use $k$ to find the new speed $v$ when $p=16$ kW and $r=0.5$ m:** $$v = 25 \times 16 \times (0.5)^2$$ Calculate $(0.5)^2$: $$0.5^2 = 0.25$$ So, $$v = 25 \times 16 \times 0.25$$ Calculate $16 \times 0.25$: $$16 \times 0.25 = 4$$ Therefore, $$v = 25 \times 4 = 100$$ 5. **Final answer:** The speed of the vehicle is $\boxed{100}$ m/s when the power is 16 kW and the radius of the wheels is 0.5 m.