Mathematical statement 2x2 - 7x - 6

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 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 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 -7. iii) Constant term: The constant term is the term without any variable. The constant term is -6. 15. Volume of a cube is given as (2445 × 0.106) 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 × 0.106 cm^3. The formula for the volume of a cube is V = s^3. So, s^3 = 2445 × 0.106. To find s, we take the cube root: s = [3]2445 × 0.106. Step 1: Take the logarithm of both sides. s = ( (2445 × 0.106)^(1)/(3) ) s = (1)/(3) (2445 × 0.106) s = (1)/(3) ( 2445 + 0.106) Step 2: Find the logarithms of 2445 and 0.106 using logarithm tables. For 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, 2445 = 3.3874 + 0.0009 = 3.3883. For 0.106: Characteristic = 1 (since the first non-zero digit is in the first decimal place). Mantissa for 10 under 6 is 0.0253. So, 0.106 = 1.0253. Step 3: Add the logarithms. (2445 × 0.106) = 3.3883 + 1.0253 = (3 + (-1)) + (0.3883 + 0.0253) = 2 + 0.4136 = 2.4136 Step 4: Divide the sum by 3. s = (1)/(3) (2.4136) 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 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 ≈ 6.4 cm. The length of one side of the cube is 6.4 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 × (3)^(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 × (3)^(t)/(5) Step 2: Divide both sides by P_0. 9 = 3^(t)/(5) Step 3: Express 9 as a power of 3. 3^2 = 3^(t)/(5) Step 4: Equate the exponents. Since the bases are the same, the exponents must be equal. 2 = (t)/(5) Step 5: Solve for t. t = 2 × 5 t = 10 It takes 10 years for the population to reach 9 times its original size.