1. **State the problem:** We have an exponential model of the form $$y = a \cdot b^x$$ that passes through points $(3,5)$ and $(4,10)$. We need to find which of the given points also lies on this graph. 2. **Write the system of equations:** Using the points, $$5 = a \cdot b^3$$ $$10 = a \cdot b^4$$ 3. **Divide the second equation by the first to eliminate $a$:** $$\frac{10}{5} = \frac{a \cdot b^4}{a \cdot b^3} \Rightarrow 2 = b^{4-3} = b$$ 4. **Substitute $b=2$ back into the first equation to find $a$:** $$5 = a \cdot 2^3 = a \cdot 8$$ $$a = \frac{5}{8}$$ 5. **Write the full equation:** $$y = \frac{5}{8} \cdot 2^x$$ 6. **Check each given point to see which satisfies the equation:** - For $(2,0)$: $$y = \frac{5}{8} \cdot 2^2 = \frac{5}{8} \cdot 4 = \frac{20}{8} = 2.5 \neq 0$$ - For $(2,1)$: $$y = \frac{5}{8} \cdot 4 = 2.5 \neq 1$$ - For $(5,15)$: $$y = \frac{5}{8} \cdot 2^5 = \frac{5}{8} \cdot 32 = \frac{160}{8} = 20 \neq 15$$ - For $(5,20)$: $$y = \frac{5}{8} \cdot 32 = 20$$ 7. **Conclusion:** The point $(5,20)$ lies on the graph. **Final answer:** $(5,20)$