A curve in the plane has the property that its slope at every point is three times the height of…

Step 1. Separate and integrate. $\dfrac{dy}{y} = 3\,dx$ gives $\ln y = 3x + c$, so $y = Ce^{3x}$.

Step 2. The constant. At $x = 0$: $5 = Ce^{0} = C$. The curve is $y = 5e^{3x}$.

Step 3. The height at $x = 0.5$. $y = 5e^{1.5} = 5 \times 4.4817 = 22.4084$.

Step 4. Where the height is $20$. $20 = 5e^{3x}$ gives $e^{3x} = 4$, so $3x = \ln 4$ and $x = \dfrac{\ln 4}{3} = 0.4621$.