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.
ScanSolve AI Answer
d) Step 1: Write the quadratic equation. x^2 - 5x - 6 = 0 Step 2: Factor the quadratic expression. Find two numbers that multiply to -6 and add to -5. These numbers are -6 and 1. (x-6)(x+1) = 0 Step 3: Set each factor equal to zero and solve for x. x-6 = 0 or x+1 = 0 x = 6 or x = -1 The solutions to the quadratic equation are x=6 or x=-1. 4a) Step 1: Determine the real-life height of the room. The scale is 1:500, meaning 1 cm on the model represents 500 cm in real life. Model length = 5 cm. Real-life length = 5 cm × 500 = 2500 cm. Convert to meters: 2500 cm = (2500)/(100) m = 25 m. The height of the room is 25 m. Step 2: Determine the corresponding area on the model. The scale for length is 1:500. The scale for area is (1:500)^2 = 1:500^2 = 1:250000. Real-life floor area = 36 m^2. Convert real-life area to cm^2: 1 m^2 = (100 cm)^2 = 10000 cm^2. Real-life floor area = 36 × 10000 cm^2 = 360000 cm^2. Area on model = Real-life area × (1)/(250000). Area on model = 360000 cm^2 × (1)/(250000) = (36)/(25) cm^2 = 1.44 cm^2. The corresponding area on the floor of the model is 1.44 cm^2. 4b) Step 1: Determine the corresponding volume on the model. The scale for length is 1:500. The scale for volume is (1:500)^3 = 1:500^3 = 1:125000000. Real-life volume = 120 m^3. Convert real-life volume to cm^3: 1 m^3 = (100 cm)^3 = 1000000 cm^3. Real-life volume = 120 × 1000000 cm^3 = 120000000 cm^3. Volume on model = Real-life volume × (1)/(125000000). Volume on model = 120000000 cm^3 × (1)/(125000000) = (120)/(125) cm^3. Step 2: Simplify the fraction. (120)/(125) = (24 × 5)/(25 × 5) = (24)/(25) = 0.96 cm^3 The corresponding volume of the model is 0.96 cm^3. 5) Step 1: Identify similar triangles. Given that BA is parallel to CE, we can conclude that CDE is similar to BDA. This is because: • DCE = DBA (corresponding angles) • CED = BAD (corresponding angles) • CDE = BDA (common angle) Step 2: Set up the ratio of corresponding sides. From the similarity CDE BDA, we have: (CE)/(BA) = (DE)/(DA) Step 3: Substitute the given values. We are given: BA = 9 cm CE = 4 cm AE = 3 cm Let DE = x. Then DA = DE + AE = x + 3. Substitute these into the ratio: (4)/(9) = (x)/(x+3) Step 4: Solve for x. Cross-multiply: 4(x+3) = 9x 4x + 12 = 9x Subtract 4x from both sides: 12 = 9x - 4x 12 = 5x Divide by 5: x = (12)/(5) x = 2.4 cm The length of DE is 2.4 cm. That's 2 down. 3 left today — send the next one.