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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You're on a roll — Step 1: Determine the height of the post. Let the height of the post be h. Let P be the point on the ground, and T be the top of the post. The distance from P to the base of the post is 20 m. The angle of elevation of T from P is 30^. Using trigonometry in the right-angled triangle formed: () = oppositeadjacent (30^) = (h)/(20) h = 20 (30^) h = 20 × (1)/(sqrt(3)) = (20)/(sqrt(3)) m Step 2: Determine the height of point S. S is the midpoint of the post, so its height from the ground is half the total height of the post. Let the height of S be h_S. h_S = (1)/(2) h = (1)/(2) × (20)/(sqrt(3)) = (10)/(sqrt(3)) m Step 3: Calculate the angle of elevation of S from P. Let the angle of elevation of S from P be . Using trigonometry in the right-angled triangle formed by P, the base of the post, and S: () = (h_S)/(20) () = 10/sqrt(3)20 () = (10)/(20sqrt(3)) = (1)/(2sqrt(3)) = ((1)/(2sqrt(3))) ≈ (0.288675) ≈ 16.102^ Rounding to 1 decimal place: ≈ 16.1^ The angle of elevation of S from P is 16.1^. a) Solve the inequalities. The given combined inequality is x - 5 3x - 8 < 2x - 3. This can be split into two separate inequalities: 1) x - 5 3x - 8 2) 3x - 8 < 2x - 3 Solve the first inequality: x - 5 3x - 8 Subtract x from both sides: -5 2x - 8 Add 8 to both sides: 3 2x Divide by 2: (3)/(2) x x 1.5 Solve the second inequality: 3x - 8 < 2x - 3 Subtract 2x from both sides: x - 8 < -3 Add 8 to both sides: x < 5 Combine the solutions from both inequalities: 1.5 x < 5 The solution to the inequalities is 1.5 x < 5. b) List the integral values of x. Integral values are integers. The integers that satisfy 1.5 x < 5 are 2, 3, and 4. The integral values of x are 2, 3, 4. Step 1: Simplify the denominator. 0.0608 + 0.0032 = 0.0640 = 0.064 Step 2: Simplify the numerator. 0.4 × 1.28 We can write these as fractions: 0.4 = (4)/(10) 1.28 = (128)/(100) 0.4 × 1.28 = (4)/(10) × (128)/(100) = (512)/(1000) Step 3: Substitute the simplified numerator and denominator back into the expression. 3/(0.0608 + 0.0032) = 3/(0.064) Convert 0.064 to a fraction: 0.064 = (64)/(1000) Now substitute this back: 3/(64/1000) When dividing by a fraction, multiply by its reciprocal: 3/(1000) × (1000)/(64) 3/(64) Step 4: Perform the division inside the cube root. 512 ÷ 64 = 8 Step 5: Calculate the cube root. [3]8 = 2 The evaluated expression is 2. What's next?