1. **Problem:** Find the approximate average annual growth rate of Calgary's population from 1971 to 2016. Step 1: Identify variables. - Initial population $P_0 = 403,319$ - Final population $P = 1,239,220$ - Time period $t = 2016 - 1971 = 45$ years Step 2: Use the exponential growth formula: $$P = P_0 (1 + r)^t$$ where $r$ is the annual growth rate. Step 3: Substitute known values: $$1,239,220 = 403,319 (1 + r)^{45}$$ Step 4: Divide both sides by 403,319: $$\frac{1,239,220}{403,319} = (1 + r)^{45}$$ $$3.071 = (1 + r)^{45}$$ Step 5: Take the 45th root: $$1 + r = 3.071^{\frac{1}{45}}$$ Step 6: Calculate: $$1 + r \approx e^{\frac{\ln(3.071)}{45}} \approx e^{0.0248} \approx 1.0251$$ Step 7: Find $r$: $$r = 1.0251 - 1 = 0.0251 = 2.51\%$$ **Answer:** Approximately 2.5% annual growth rate (Option B). 2. **Problem:** Find the half-life of a drug given initial and one-day plasma levels. Step 1: Identify variables. - Initial level $P_0 = 3600$ mg/L - Level after 1 day $P = 1160$ mg/L - Time $t = 1$ day Step 2: Use exponential decay formula: $$P = P_0 \left(\frac{1}{2}\right)^{\frac{t}{h}}$$ where $h$ is the half-life in days. Step 3: Substitute known values: $$1160 = 3600 \left(\frac{1}{2}\right)^{\frac{1}{h}}$$ Step 4: Divide both sides by 3600: $$\frac{1160}{3600} = \left(\frac{1}{2}\right)^{\frac{1}{h}}$$ $$0.3222 = \left(\frac{1}{2}\right)^{\frac{1}{h}}$$ Step 5: Take natural log: $$\ln(0.3222) = \frac{1}{h} \ln\left(\frac{1}{2}\right)$$ Step 6: Calculate: $$-1.132 = \frac{1}{h} (-0.693)$$ Step 7: Solve for $h$: $$\frac{1}{h} = \frac{-1.132}{-0.693} = 1.633$$ $$h = \frac{1}{1.633} = 0.612 \text{ days}$$ Step 8: Convert days to hours: $$0.612 \times 24 = 14.7 \text{ hours}$$ **Answer:** Half-life is approximately 14.7 hours (Option B). 3. **Problem:** Find how many additional years Harry must wait to double his investment. Step 1: Identify variables. - Initial investment $P_0 = 1000$ - Value after 2 years $P = 1127.84$ - Goal $P_g = 2000$ - Time elapsed $t_1 = 2$ years - Additional time $t_2$ to find Step 2: Find annual growth rate $r$ using: $$P = P_0 (1 + r)^{t_1}$$ Step 3: Substitute values: $$1127.84 = 1000 (1 + r)^2$$ Step 4: Divide both sides: $$1.12784 = (1 + r)^2$$ Step 5: Take square root: $$1 + r = \sqrt{1.12784} = 1.0619$$ Step 6: Calculate $r$: $$r = 0.0619 = 6.19\%$$ Step 7: Use formula to find total time $T$ to reach $2000$: $$2000 = 1000 (1.0619)^T$$ Step 8: Divide: $$2 = (1.0619)^T$$ Step 9: Take natural log: $$\ln(2) = T \ln(1.0619)$$ Step 10: Calculate: $$0.693 = T \times 0.0601$$ Step 11: Solve for $T$: $$T = \frac{0.693}{0.0601} = 11.53 \text{ years}$$ Step 12: Find additional years: $$t_2 = T - t_1 = 11.53 - 2 = 9.53 \approx 9.5 \text{ years}$$ **Answer:** Harry must wait approximately 9.5 more years (Option D). 4. **Problem:** Solve algebraically for $x$ in $$\frac{5^{x^2 + x}}{125^{x-1}} = 5 \left(\frac{1}{25}\right)^{x-2}$$ Step 1: Express all bases as powers of 5: - $125 = 5^3$ - $\frac{1}{25} = 5^{-2}$ Step 2: Rewrite equation: $$\frac{5^{x^2 + x}}{(5^3)^{x-1}} = 5 \times (5^{-2})^{x-2}$$ Step 3: Simplify exponents: $$\frac{5^{x^2 + x}}{5^{3(x-1)}} = 5 \times 5^{-2(x-2)}$$ Step 4: Use quotient rule: $$5^{x^2 + x - 3x + 3} = 5^{1 - 2x + 4}$$ Step 5: Simplify exponents: $$5^{x^2 - 2x + 3} = 5^{5 - 2x}$$ Step 6: Since bases are equal, set exponents equal: $$x^2 - 2x + 3 = 5 - 2x$$ Step 7: Simplify: $$x^2 - 2x + 3 = 5 - 2x$$ $$x^2 - 2x + 3 - 5 + 2x = 0$$ $$x^2 - 2 = 0$$ Step 8: Solve for $x$: $$x^2 = 2$$ $$x = \pm \sqrt{2}$$ **Answer:** $x = \sqrt{2}$ or $x = -\sqrt{2}$. 5. **Problem:** Termite population doubles every 20 days starting from 1200. Find time $t$ to reach 153600. Step 1: Use exponential growth formula: $$P = P_0 2^{\frac{t}{d}}$$ where $d=20$ days. Step 2: Substitute values: $$153600 = 1200 \times 2^{\frac{t}{20}}$$ Step 3: Divide both sides: $$128 = 2^{\frac{t}{20}}$$ Step 4: Express 128 as power of 2: $$128 = 2^7$$ Step 5: Equate exponents: $$2^7 = 2^{\frac{t}{20}} \Rightarrow 7 = \frac{t}{20}$$ Step 6: Solve for $t$: $$t = 7 \times 20 = 140 \text{ days}$$ **Answer:** It will take 140 days for the termite population to reach approximately 153600.