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 answers to your questions: GSE 212: BASIC GENERAL MATHEMATICS III (1C) Question 1: a) Define the following terms: (i) Measurement: The process of assigning a numerical value to a physical quantity using a standard unit. (ii) Quantity: A property of a phenomenon, body, or substance that can be quantified by measurement. (iii) Unit: A standard amount of a physical quantity, such as a meter for length or a kilogram for mass, used for comparison. (iv) Accuracy: How close a measured value is to the true or actual value of the quantity being measured. (v) Precision: The degree to which repeated measurements under unchanged conditions show the same results; it indicates the reproducibility of a measurement. b) Given 3x + 4y = 26, express x in terms of y and find x when y = 5 and y = 10. Step 1: Express x in terms of y. Subtract 4y from both sides, then divide by 3. 3x + 4y = 26 3x = 26 - 4y x = (26 - 4y)/(3) The expression for x in terms of y is x = (26 - 4y)/(3). Step 2: Find x when y = 5. Substitute y = 5 into the expression for x. x = (26 - 4(5))/(3) x = (26 - 20)/(3) x = (6)/(3) x = 2 When y = 5, x = 2. Step 3: Find x when y = 10. Substitute y = 10 into the expression for x. x = (26 - 4(10))/(3) x = (26 - 40)/(3) x = (-14)/(3) When y = 10, x = -(14)/(3). c) In the diagram, PQST is a parallelogram, PR is a straight line. |TS| = 8 cm, |SM| = 6 cm and area of triangle PSR = 36 cm^2. Find the value of |QR|. Step 1: Find the length of the base PR using the area of triangle PSR. The area of a triangle is given by (1)/(2) × base × height. Here, the base is PR and the height is SM. Area of PSR = (1)/(2) × |PR| × |SM| 36 cm^2 = (1)/(2) × |PR| × 6 cm 36 = 3 × |PR| |PR| = (36)/(3) |PR| = 12 cm Step 2: Determine the length of PQ. Since PQST is a parallelogram, opposite sides are equal in length. Therefore, |PQ| = |TS|. |PQ| = 8 cm Step 3: Calculate the value of |QR|. From the diagram, PR is a straight line, so |PR| = |PQ| + |QR|. 12 cm = 8 cm + |QR| |QR| = 12 - 8 |QR| = 4 cm The value of |QR| is 4 cm. Question 2: a) The formula T = (4b^2)/(21)(d - (3)/(5)b) gives the approximate mass, T tonnes, of a ship d metres long and b metres wide. i) Make d the subject of the formula. Step 1: Multiply both sides by 21. 21T = 4b^2 (d - (3)/(5)b) Step 2: Divide both sides by 4b^2. (21T)/(4b^2) = d - (3)/(5)b Step 3: Add (3)/(5)b to both sides. d = (21T)/(4b^2) + (3)/(5)b The formula with d as the subject is d = (21T)/(4b^2) + (3)/(5)b. ii) Calculate the length of a 4500 tonnes ship if it is 20m wide, giving your answer to the nearest 5m. Step 1: Substitute the given values into the formula for d. Given T = 4500 tonnes and b = 20 m. d = (21(4500))/(4(20)^2) + (3)/(5)(20) d = (94500)/(4(400)) + (60)/(5) d = (94500)/(1600) + 12 Step 2: Perform the calculations. d = 59.0625 + 12 d = 71.0625 m Step 3: Round the answer to the nearest 5m. 71.0625 m is closer to 70 m than 75 m. The length of the ship is approximately 70 m. b) A wooden structure in form of a cone mounted on a hemispherical base. The vertical height of the cone is 24cm and the base radius 7cm. Calculate, correct to 3 Significant figures, the surface area of the structure. ( = (22)/(7)). Step 1: Calculate the slant height (l) of the cone. The slant height can be found using the Pythagorean theorem: l = sqrt(r^2 + h^2). l = sqrt(7^2 + 24^2) l = sqrt(49 + 576) l = sqrt(625) l = 25 cm Step 2: Calculate the curved surface area of the cone (CSA_cone). The formula for the curved surface area of a cone is r l. CSA_cone = (22)/(7) × 7 cm × 25 cm CSA_cone = 22 × 25 cm^2 CSA_cone = 550 cm^2 Step 3: Calculate the curved surface area of the hemisphere (CSA_hemi). The formula for the curved surface area of a hemisphere is 2 r^2. CSA_hemi = 2 × (22)/(7) × (7 cm)^2 CSA_hemi = 2 × (22)/(7) × 49 cm^2 CSA_hemi = 2 × 22 × 7 cm^2 CSA_hemi = 308 cm^2 Step 4: Calculate the total surface area of the structure. The total surface area is the sum of the curved surface area of the cone and the curved surface area of the hemisphere (the base of the cone is covered by the hemisphere). Total Surface Area = CSA_cone + CSA_hemi Total Surface Area = 550 cm^2 + 308 cm^2 Total Surface Area = 858 cm^2 Step 5: Express the answer correct to 3 significant figures. 858 cm^2 is already expressed to 3 significant figures. The surface area of the structure is 858 cm^2. 3 done, 2 left today. You're making progress.