The rectangles A and B have perimeters of 94cm and 56cm as shown below (insert diagram). Rectangle A: base = 2x cm, height = 3y cm. Rectangle B: base = (x+6)cm, height = (y+4)cm. Use an algebraic method to calculate the area of each rectangle. (8 marks)
Formulas needed: Perimeter of rectangle = 2*(base + height) Area of rectangle = base * height. Using the information given we need to define the perimeter of A and B algebraically. Rectangle A: base = 2x, height = 3y, Perimeter = 94cm. Sub into Perimeter formula: 94 = 2(2x+3y) 94= 4x+6y. This gives us simultaneous equation 1. Rectangle B: base = x+6, height = y+4, Perimeter = 56cm. Sub into Perimeter formula: 56 = 2((x+6)+(y+4)) 56 = 2(x+6) + 2(y+4) 56 = 2x+12 + 2y+8. Simplify by collecting number terms and algebraic terms 56 = 2x + 2y + 20. Take +20 over to the other side to become -20. 56-20 = 2x + 2y 36 = 2x + 2y. This gives us simultaneous equation 2. 94 = 4x+6y. eqn 1. 36 = 2x + 2y. eqn 2. We have 2 simultaneous equations with 2 unknowns. To simplify we will double all the terms in equation 2. We will then have 4x in both equations. This allows us to subtract eqn 1 from eqn 2 and erases the x term. Equation 2 becomes: 72 = 4x + 4y. We will call this equation 2" Now: 94 = 4x+6y: eqn 1. 72 = 4x+4y: eqn 2". Eqn 1 - Eqn 2"(Subtract all the like terms): (94-72) = (4x-4x)+(6y-4y) 22 = 0x + 2y 22 = 2y y = 22/2 y = 11Sub y=11 into eqn 2: (We use eqn 2 rather than eqn 2" because the numbers are smaller, which it makes it easier to calculate in a non-calculator paper) 36 = 2x + 2(11) 36 = 2x + 22 36-22 = 2x 14 = 2x x = 14/2 x = 7y=11, x=7 Now that x and y are found we can substitute them into the dimensions for base and height. Then we can calculate the total area of each rectangle.Rectangle A:base = 2x cm = 2(7) = 14cmheight = 3y cm = 3(11) = 33cm Use formula for Area of rectangle. Area = base * height = 14 * 33 14 * 33 can be written as (1033)+(433) Area = 330 + 132 = 462 Area rectangle A = 462 cm2Rectangle B: base = (x+6)cm = 7+6 = 13cm height = (y+4)cm = 11+4 = 15cm Area = baseheight = 1315 1315 can be written as (1015)+(3*15) Area = 150 + 45 = 195 Area rectangle B = 195 cm2
