Here are the solutions to the problems: 14. Mathematical statement $2x^2 - 7x - 6$ a) Name given to this type of mathematical statement: This is a polynomial of degree 2. The name is a quadratic expression or quadratic polynomial. The name is $\boxed{\text{Quadratic expression}}$. b) From the mathematical statement, write down the: i) Quadratic term: The term containing $x^2$ is $2x^2$. The quadratic term is $\boxed{2x^2}$. ii) Coefficient of the linear term: The linear term is the term containing $x$, which is $-7x$. The coefficient is the number multiplying $x$. The coefficient of the linear term is $\boxed{-7}$. iii) Constant term: The constant term is the term without any variable. The constant term is $\boxed{-6}$. 15. Volume of a cube is given as $(2445 \times 0.106) \text{ cm}^3$. Use common logarithm tables to calculate the length of one side of the cube. Give your answer correct to 1 d.p. Let $V$ be the volume of the cube and $s$ be the length of one side. We are given $V = 2445 \times 0.106 \text{ cm}^3$. The formula for the volume of a cube is $V = s^3$. So, $s^3 = 2445 \times 0.106$. To find $s$, we take the cube root: $s = \sqrt[3]{2445 \times 0.106}$. Step 1: Take the logarithm of both sides. $$\log s = \log \left( (2445 \times 0.106)^{\frac{1}{3}} \right)$$ $$\log s = \frac{1}{3} \log (2445 \times 0.106)$$ $$\log s = \frac{1}{3} (\log 2445 + \log 0.106)$$ Step 2: Find the logarithms of $2445$ and $0.106$ using logarithm tables. For $\log 2445$: Characteristic = $3$ (since $2445$ has 4 digits, $4-1=3$). Mantissa for $24$ under $4$ is $0.3874$. Mean difference for $5$ is $9$. So, $\log 2445 = 3.3874 + 0.0009 = 3.3883$. For $\log 0.106$: Characteristic = $\bar{1}$ (since the first non-zero digit is in the first decimal place). Mantissa for $10$ under $6$ is $0.0253$. So, $\log 0.106 = \bar{1}.0253$. Step 3: Add the logarithms. $$\log (2445 \times 0.106) = 3.3883 + \bar{1}.0253$$ $$ = (3 + (-1)) + (0.3883 + 0.0253)$$ $$ = 2 + 0.4136$$ $$ = 2.4136$$ Step 4: Divide the sum by 3. $$\log s = \frac{1}{3} (2.4136)$$ $$\log s = 0.804533...$$ We will use $0.8045$ for finding the antilog. Step 5: Find the antilog of $0.8045$. Antilog of $0.8045$: Look up $0.80$ under $4$, which gives $6.368$. The mean difference for $5$ (in the row for $80$) is $7$. So, antilog $0.8045 = 6.368 + 0.007 = 6.375$. Since the characteristic of $\log s$ is $0$ (from $0.8045$), the decimal point is after the first digit. So, $s = 6.375$. Step 6: Round the answer to 1 decimal place. $s \approx 6.4$ cm. The length of one side of the cube is $\boxed{6.4 \text{ cm}}$. 16. The population of a small village in Turkana grows at a rate modeled using indices. If the initial population $P$ triples every 5 years, use laws of indices to find how long it takes for the population to reach 9 times its original size. Let the initial population be $P_0$. The population triples every 5 years. After $t$ years, the population $P(t)$ can be expressed as: $$P(t) = P_0 \times (3)^{\frac{t}{5}}$$ We want to find the time $t$ when the population reaches 9 times its original size, i.e., $P(t) = 9P_0$. Step 1: Set up the equation. $$9P_0 = P_0 \times (3)^{\frac{t}{5}}$$ Step 2: Divide both sides by $P_0$. $$9 = 3^{\frac{t}{5}}$$ Step 3: Express 9 as a power of 3. $$3^2 = 3^{\frac{t}{5}}$$ Step 4: Equate the exponents. Since the bases are the same, the exponents must be equal. $$2 = \frac{t}{5}$$ Step 5: Solve for $t$. $$t = 2 \times 5$$ $$t = 10$$ It takes $\boxed{10 \text{ years}}$ for the population to reach 9 times its original size.