1. **Problem 16:** A chemist forms crystals that double every 24 minutes starting with 4 crystals. Find the equation modeling the number of crystals $n$ after $t$ hours. 2. The doubling time is 24 minutes, but $t$ is in hours. Convert hours to minutes: $t$ hours $= 60t$ minutes. 3. The number of crystals doubles every 24 minutes, so the number of doubling periods in $t$ hours is $\frac{60t}{24} = \frac{5t}{2}$. 4. The general exponential growth formula is $$n = n_0 \times 2^{\text{number of doubling periods}}$$ where $n_0=4$. 5. Substitute: $$n = 4 \times 2^{\frac{5t}{2}}$$ which matches option B. --- 1. **Problem 17:** Identify which withdrawal plan yields exponential decay. 2. Exponential decay means the amount decreases by a fixed percentage of the current balance each year. 3. Option A withdraws a fixed percent of the initial balance (not current), so not exponential decay. 4. Option B withdraws a fixed percent of initial plus fixed amount, not exponential decay. 5. Option C withdraws 3% of the current value each year, which is exponential decay. 6. Option D withdraws a fixed amount, not exponential decay. 7. Therefore, option C yields exponential decay. --- 1. **Problem 18:** Find $x$ in $180x^t$ for 3% interest compounded annually. 2. Compound interest formula: $$A = P(1 + r)^t$$ where $r=0.03$. 3. So, $x = 1 + 0.03 = 1.03$. --- 1. **Problem 19:** Compare Hayden's and Matilda's accounts after 8 years. 2. Matilda's amount: $$180 \times 1.03^8$$ 3. Hayden's amount: $$180 \times 1.035^8$$ 4. Calculate each: $$1.03^8 \approx 1.26677$$ $$1.035^8 \approx 1.31607$$ 5. Matilda's amount: $$180 \times 1.26677 = 228.02$$ 6. Hayden's amount: $$180 \times 1.31607 = 236.89$$ 7. Difference: $$236.89 - 228.02 = 8.87$$ **Final answers:** - Problem 16: Option B: $$n = 4 \times 2^{\frac{5t}{2}}$$ - Problem 17: Option C - Problem 18: $x = 1.03$ - Problem 19: Hayden earns 8.87 more than Matilda after 8 years.