1. The problem states that the function $f$ is exponential and passes through points $(0,5)$ and $(1.5,10)$. We need to find $f(3)$. 2. The general form of an exponential function is $$f(x) = ab^x$$ where $a$ is the initial value and $b$ is the base or growth factor. 3. Since the graph passes through $(0,5)$, we know $$f(0) = ab^0 = a = 5.$$ So, $a=5$. 4. Using the point $(1.5,10)$, substitute into the function: $$10 = 5b^{1.5}.$$ Divide both sides by 5: $$\frac{10}{5} = \cancel{5}b^{1.5} / \cancel{5} \Rightarrow 2 = b^{1.5}.$$ 5. To solve for $b$, raise both sides to the power of $\frac{1}{1.5} = \frac{2}{3}$: $$b = 2^{\frac{2}{3}} = \left(2^{\frac{1}{3}}\right)^2 = \sqrt[3]{2}^2.$$ 6. Now find $f(3)$: $$f(3) = 5b^3 = 5 \times \left(2^{\frac{2}{3}}\right)^3 = 5 \times 2^{2} = 5 \times 4 = 20.$$ Final answer: $$\boxed{20}.$$