1. **Problem statement:** a) Find the exponential function $k$ with initial value 4 and doubling time 10 years. b) Find the exponential function $k$ with initial value 12.8 and half-life 17 days. 2. **Formula for exponential growth/decay:** The general form is $$k(t) = k_0 \cdot a^{\frac{t}{T}}$$ where $k_0$ is the initial value, $a$ is the growth/decay factor, $t$ is time, and $T$ is the doubling or half-life period. 3. **Important rules:** - For doubling time $T$, the growth factor $a = 2$. - For half-life $T$, the decay factor $a = \frac{1}{2}$. 4. **Part a) Doubling time 10 years, initial value 4:** Using the formula: $$k(t) = 4 \cdot 2^{\frac{t}{10}}$$ 5. **Part b) Half-life 17 days, initial value 12.8:** Using the formula: $$k(t) = 12.8 \cdot \left(\frac{1}{2}\right)^{\frac{t}{17}}$$ 6. **Explanation:** - In part a), the function doubles every 10 years, so the base of the exponent is 2. - In part b), the function halves every 17 days, so the base of the exponent is $\frac{1}{2}$. **Final answers:** $$\boxed{\text{a) } k(t) = 4 \cdot 2^{\frac{t}{10}}}$$ $$\boxed{\text{b) } k(t) = 12.8 \cdot \left(\frac{1}{2}\right)^{\frac{t}{17}}}$$