This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to question 2: 2. MATHEMATICS a) If y varies inversely as sqrt(x), and x is multiplied by n. What is the ratio of the first y to the second y? Step 1: Express the inverse variation relationship. If y varies inversely as sqrt(x), then y = (k)/(sqrt(x)), where k is the constant of proportionality. Let the first y be y_1, so y_1 = (k)/(sqrt(x)). Step 2: Determine the second y when x is multiplied by n. The new value of x is nx. Let the second y be y_2. So, y_2 = (k)/(sqrt(nx)). Step 3: Find the ratio of the first y to the second y. (y_1)/(y_2) = (k)/(sqrt(x))(k)/(sqrt(nx)) Step 4: Simplify the ratio. (y_1)/(y_2) = (k)/(sqrt(x)) × sqrt(nx)k (y_1)/(y_2) = sqrt(nx)sqrt(x) (y_1)/(y_2) = sqrt((nx)/(x)) (y_1)/(y_2) = sqrt(n) The ratio of the first y to the second y is sqrt(n). b) The headmaster has enough food to last for his 600 students for 20 days from tomorrow. If 120 students leave the school today for UMISSETA game, how long will the food last? Step 1: Identify the initial conditions. Number of students (S_1) = 600 Number of days food lasts (D_1) = 20 days Step 2: Calculate the new number of students. 120 students leave, so the new number of students (S_2) is: S_2 = 600 - 120 = 480 students. Step 3: Apply the inverse proportion relationship. The total amount of food is constant, so S_1 × D_1 = S_2 × D_2. 600 students × 20 days = 480 students × D_2 days 12000 = 480 D_2 Step 4: Solve for D_2. D_2 = (12000)/(480) D_2 = (1200)/(48) D_2 = 25 The food will last for 25 days. c) A regular Hexagon is inscribed in a circle if the perimeter of the hexagon is 42cm, find the radius of the circle and its Area Step 1: Determine the side length of the hexagon. A regular hexagon has 6 equal sides. Perimeter of hexagon = 6 × side length. Given perimeter = 42 cm. 6 × side length = 42 cm side length = (42)/(6) = 7 cm Step 2: Find the radius of the circle. For a regular hexagon inscribed in a circle, the side length of the hexagon is equal to the radius of the circle. Therefore, the radius of the circle (r) is 7 cm. Radius of the circle = 7 cm. Step 3: Calculate the area of the circle. The area of a circle is given by the formula A = r^2. Using r = 7 cm: A = (7 cm)^2 A = 49 cm^2 If using ≈ (22)/(7): A = (22)/(7) × (7 cm)^2 A = (22)/(7) × 49 cm^2 A = 22 × 7 cm^2 A = 154 cm^2 The area of the circle is 49 cm^2 (or 154 cm^2 if = (22)/(7) is used).