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 solutions to the trigonometry problems: Problem 1: Ladder against a wall Step 1: Identify the knowns and unknowns. The ladder, the wall, and the ground form a right-angled triangle. • Length of the ladder (hypotenuse) = 13 m • Distance from the base of the wall to the foot of the ladder (adjacent side) = 5 m • Height the ladder reaches up the wall (opposite side) = h (unknown) Step 2: Apply the Pythagorean theorem. The Pythagorean theorem states a^2 + b^2 = c^2, where c is the hypotenuse and a and b are the other two sides. h^2 + 5^2 = 13^2 Step 3: Solve for h. h^2 + 25 = 169 h^2 = 169 - 25 h^2 = 144 h = sqrt(144) h = 12 The ladder reaches 12 m up the wall. Problem 2: Angle of depression of a boat Step 1: Identify the knowns and unknowns. The cliff, the sea level, and the line of sight to the boat form a right-angled triangle. The angle of depression from the top of the cliff to the boat is equal to the angle of elevation from the boat to the top of the cliff. • Height of the cliff (opposite side to the angle of elevation) = 15 m • Angle of depression = 35^ (which is also the angle of elevation from the boat) • Distance from the base of the cliff to the boat (adjacent side) = x (unknown) Step 2: Use the tangent function. The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side: () = oppositeadjacent (35^) = (15)/(x) Step 3: Solve for x. We are given 35^ = 0.7. 0.7 = (15)/(x) x = (15)/(0.7) x ≈ 21.42857 Step 4: Round to two decimal places. x ≈ 21.43 The distance from the base of the cliff to the boat is 21.43 m. Problem 3: Angle of depression to a swimmer Step 1: Identify the knowns and unknowns. The cliff, the sea level, and the line of sight to the swimmer form a right-angled triangle. • Height of the cliff (opposite side) = 30 m • Horizontal distance to the swimmer (adjacent side) = 15 m • Angle of depression = (unknown) Step 2: Use the tangent function. () = oppositeadjacent () = (30)/(15) () = 2 Step 3: Determine the angle using the given values. We are given: 64^ = 2.0503 65^ = 2.1444 66^ = 2.2460 Since () = 2, the angle is between 63^ and 64^. Based on the provided options, the closest integer angle would be 63^ if we were to round, but since 2 is not exactly one of the given values, we can state the exact value of . = (2) Using a calculator, ≈ 63.43^. However, if we are to choose from the given values, none of them exactly match () = 2. If the question implies finding the closest integer angle from the given table, then 2 is closest to 2.0503 (for 64^) than to 2.1444 (for 65^). But 2 is also less than 2.0503. Let's assume the question expects us to find the angle whose tangent is exactly 2. The angle of depression is (2) ≈ 63.43^. Problem 4: Height of a tree Step 1: Identify the knowns and unknowns. We have a right-angled triangle formed by Bill's eye level, the horizontal distance to the tree, and the height from Bill's eye level to the top of the tree. • Bill's height = 1.5 m • Horizontal distance from Bill to the tree (adjacent side) = 5 m • Angle of elevation = 20^ • Height from Bill's eye level to the top of the tree = h' (unknown) Step 2: Use the tangent function to find h'. () = oppositeadjacent (20^) = (h')/(5) Step 3: Solve for h'. We are given 20^ = 0.36. 0.36 = (h')/(5) h' = 0.36 × 5 h' = 1.8 m Step 4: Calculate the total height of the tree. The total height of the tree is Bill's height plus h'. Total height = Bill's height + h' Total height = 1.5 m + 1.8 m Total height = 3.3 m The height of the tree is 3.3 m. 3 done, 2 left today. You're making progress.