Simplify: 1.a) (34 x 3^-2)/33, 1.b) Express 0.00032 in standard form, 2.a) Expand and simplify: (2x-3)(x+5)-(x-4)2

Welcome back — been a while! Let's pick up where you left off. 1.a) Simplify: We use the exponent rules a^m × a^n = a^m+n and (a^m)/(a^n) = a^m-n. 3^4 × 3^-23^3 Step 1: Simplify the numerator. 3^4 + (-2) = 3^4 - 2 = 3^2 Step 2: Divide the result by the denominator. (3^2)/(3^3) = 3^2 - 3 = 3^-1 Step 3: Express with a positive exponent. 3^-1 = (1)/(3^1) = (1)/(3) The simplified expression is (1)/(3). 1.b) Express 0.00032 in standard form: To express a number in standard form, we move the decimal point until there is one non-zero digit to its left. Step 1: Move the decimal point to the right until it is after the first non-zero digit (3). 0.00032 → 3.2 Step 2: Count the number of places the decimal point was moved. It was moved 4 places to the right. Step 3: Since the original number was less than 1, the exponent of 10 will be negative. 3.2 × 10^-4 The number in standard form is 3.2 × 10^-4. 2.a) Expand and simplify: (2x - 3)(x + 5) - (x - 4)^2 Step 1: Expand the first product (2x - 3)(x + 5) using FOIL. (2x)(x) + (2x)(5) + (-3)(x) + (-3)(5) = 2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15 Step 2: Expand the second term (x - 4)^2 using the formula (a-b)^2 = a^2 - 2ab + b^2. (x - 4)^2 = x^2 - 2(x)(4) + 4^2 = x^2 - 8x + 16 Step 3: Substitute the expanded terms back into the original expression and simplify. Remember to distribute the negative sign to all terms in the second expansion. (2x^2 + 7x - 15) - (x^2 - 8x + 16) 2x^2 + 7x - 15 - x^2 + 8x - 16 Step 4: Combine like terms. (2x^2 - x^2) + (7x + 8x) + (-15 - 16) x^2 + 15x - 31 The expanded and simplified expression is x^2 + 15x - 31. 2.b) Factorise completely: 3x^2 - 11x - 4 We are looking for two numbers that multiply to 3 × (-4) = -12 and add up to -11. These numbers are -12 and 1. Step 1: Rewrite the middle term using these two numbers. 3x^2 - 12x + x - 4 Step 2: Factor by grouping. (3x^2 - 12x) + (x - 4) 3x(x - 4) + 1(x - 4) Step 3: Factor out the common binomial factor (x - 4). (x - 4)(3x + 1) The completely factorised expression is (x - 4)(3x + 1). 2.c) Solve: 4x^2 - 9 = 0 This is a difference of squares, a^2 - b^2 = (a-b)(a+b), where a = sqrt(4x^2) = 2x and b = sqrt(9) = 3. Step 1: Factor the expression. (2x - 3)(2x + 3) = 0 Step 2: Set each factor equal to zero and solve for x. 2x - 3 = 0 or 2x + 3 = 0 2x = 3 or 2x = -3 x = (3)/(2) or x = -(3)/(2) The solutions are x = (3)/(2) or x = -(3)/(2). 3. Solve: 3x + 2y = 17 (Equation 1) 2x - y = 4 (Equation 2) We will use the elimination method. Step 1: Multiply Equation 2 by 2 to make the coefficients of y opposites. 2 × (2x - y) = 2 × 4 4x - 2y = 8 (Equation 3) Step 2: Add Equation 1 and Equation 3. (3x + 2y) + (4x - 2y) = 17 + 8 7x = 25 Step 3: Solve for x. x = (25)/(7) Step 4: Substitute the value of x into Equation 2 to solve for y. 2((25)/(7)) - y = 4 (50)/(7) - y = 4 -y = 4 - (50)/(7) -y = (28)/(7) - (50)/(7) -y = -(22)/(7) y = (22)/(7) The solutions are x = (25)/(7), y = (22)/(7). 4. Two similar cones have radii 3 cm and 9 cm. Let r_1 = 3 cm (smaller cone) and r_2 = 9 cm (larger cone). 4.a) Find the scale factor. The scale factor k is the ratio of corresponding linear dimensions. We'll use the ratio of the larger radius to the smaller radius. k = (r_2)/(r_1) = 9 cm3 cm = 3 The scale factor is 3. 4.b) If the smaller cone has volume 40 cm^3, find the volume of the larger cone. For similar solids, the ratio of their volumes is the cube of the scale factor. Let V_1 be the volume of the smaller cone and V_2 be the volume of the larger cone. (V_2)/(V_1) = k^3 (V_2)/(40 cm)^3 = 3^3 (V_2)/(40 cm)^3 = 27 V_2 = 27 × 40 cm^3 V_2 = 1080 cm^3 The volume of the larger cone is 1080 cm^3. 4.c) If the slant height of the smaller cone is 5 cm, find that of the larger cone. For similar solids, the ratio of their corresponding linear dimensions (like slant height) is equal to the scale factor. Let l_1 be the slant height of the smaller cone and l_2 be the slant height of the larger cone. (l_2)/(l_1) = k (l_2)/(5 cm) = 3 l_2 = 3 × 5 cm l_2 = 15 cm The slant height of the larger cone is 15 cm. 5. A cylindrical water tank has radius 7 m and height 10 m. Given: r = 7 m, h = 10 m. 5.a) Find its volume. The formula for the volume of a cylinder is V = r^2 h. V = (7 m)^2 (10 m) V = (49 m^2) (10 m) V = 490 m^3 The volume of the tank is 490 m^3. 5.b) Find its total surface area. The formula for the total surface area of a cylinder is A = 2 r h + 2 r^2. A = 2 (7 m)(10 m) + 2 (7 m)^2 A = 140 m^2 + 2 (49 m^2) A = 140 m^2 + 98 m^2 A = 238 m^2 The total surface area of the tank is 238 m^2. 5.c) If water costs KSh 50 per cubic meter, find the cost to fill the tank. Cost to fill the tank = Volume × Cost per cubic meter. Using the volume from 5.a): Cost = (490 m^3) × (KSh 50/m^3) Cost = 490 × 50 × KSh Cost = 24500 KSh If we use ≈ 3.14159: Cost ≈ 24500 × 3.14159 KSh Cost ≈ 76969.355 KSh The cost to fill the tank is 24500 KSh (or approximately KSh 76969.36). Send me the next one 📸