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
Here's how to solve the problem: Step 1: Find the distance between the bottom edges of the boards (side AB). We can use the Cosine Rule for triangle ABC. The Cosine Rule states c^2 = a^2 + b^2 - 2ab C. Here, c = AB, a = BC = 5\,m, b = AC = 6\,m, and C = 30^. AB^2 = AC^2 + BC^2 - 2(AC)(BC) ( C) AB^2 = 6^2 + 5^2 - 2(6)(5) (30^) AB^2 = 36 + 25 - 60 (sqrt(3)2) AB^2 = 61 - 30sqrt(3) AB^2 ≈ 61 - 30(1.73205) AB^2 ≈ 61 - 51.9615 AB^2 ≈ 9.0385 AB = sqrt(9.0385) AB ≈ 3.006409\,m Correct to three significant figures, the distance AB is: a) 3.01\,m Step 2: Find the angle the longer board makes with the floor (angle ). The longer board is AC (6m), and the angle it makes with the floor is A (labeled in the diagram). We can use the Sine Rule: (a)/( A) = (c)/( C). Here, a = BC = 5\,m, c = AB ≈ 3.006409\,m, and C = 30^. (BC)/( A) = (AB)/( C) (5)/( ) = (3.006409)/( 30^) = (5 30^)/(3.006409) = (5 × 0.5)/(3.006409) = (2.5)/(3.006409) ≈ 0.83155 = (0.83155) ≈ 56.25^ Rounding to one decimal place, the angle is: b) 56.3^ Step 3: Find the shortest distance between the top edges of the boards and the floor. This is the height (h) from point C to the base AB. We can find the area of the triangle using two different formulas and equate them. Area of ABC = (1)/(2) AC · BC C Area = (1)/(2) (6)(5) 30^ Area = (1)/(2) (30) (0.5) Area = 7.5\,m^2 Also, Area of ABC = (1)/(2) × base × height = (1)/(2) AB · h. 7.5 = (1)/(2) (3.006409) h h = (2 × 7.5)/(3.006409) h = (15)/(3.006409) h ≈ 4.9893\,m Correct to two significant figures, the shortest distance is: c) 5.0\,m That's 2 down. 3 left today — send the next one.