The manufacturer of a reclining lawn chair would like to have the chair positioned at the following angles: 105,125,145,165 and 175.
Sides AC are 75cm
AB are 55 cm
Determine length of BC which the ^ angles
To determine the length of BC, we can use the law of cosines. The law of cosines states that in a triangle with sides a, b, and c, and the angle opposite side c is denoted as angle C, the following equation holds true:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we know that sides AC and AB are 75 cm and 55 cm, respectively. We need to calculate the length of side BC for each of the given angles.
Let's calculate the length of BC for each angle:
For angle 105 degrees:
c^2 = (75)^2 + (55)^2 - 2 * 75 * 55 * cos(105 degrees)
c^2 = 5625 + 3025 - 2 * 75 * 55 * (-0.34202)
c^2 ≈ 5625 + 3025 + 4668.9
c^2 ≈ 13318.9
c ≈ √13318.9
c ≈ 115.39 cm (rounded to two decimal places)
For angle 125 degrees:
c^2 = (75)^2 + (55)^2 - 2 * 75 * 55 * cos(125 degrees)
c^2 = 5625 + 3025 - 2 * 75 * 55 * (-0.57358)
c^2 ≈ 5625 + 3025 + 5313.75
c^2 ≈ 13963.75
c ≈ √13963.75
c ≈ 118.19 cm (rounded to two decimal places)
For angle 145 degrees:
c^2 = (75)^2 + (55)^2 - 2 * 75 * 55 * cos(145 degrees)
c^2 = 5625 + 3025 - 2 * 75 * 55 * (-0.76604)
c^2 ≈ 5625 + 3025 + 5536.5
c^2 ≈ 14186.5
c ≈ √14186.5
c ≈ 119.14 cm (rounded to two decimal places)
For angle 165 degrees:
c^2 = (75)^2 + (55)^2 - 2 * 75 * 55 * cos(165 degrees)
c^2 = 5625 + 3025 - 2 * 75 * 55 * (-0.93969)
c^2 ≈ 5625 + 3025 + 6240.94
c^2 ≈ 14890.94
c ≈ √14890.94
c ≈ 122.06 cm (rounded to two decimal places)
For angle 175 degrees:
c^2 = (75)^2 + (55)^2 - 2 * 75 * 55 * cos(175 degrees)
c^2 = 5625 + 3025 - 2 * 75 * 55 * (-0.99619)
c^2 ≈ 5625 + 3025 + 6490.63
c^2 ≈ 15140.63
c ≈ √15140.63
c ≈ 123.04 cm (rounded to two decimal places)
Therefore, the lengths of side BC for the given angles are approximately:
105 degrees: 115.39 cm
125 degrees: 118.19 cm
145 degrees: 119.14 cm
165 degrees: 122.06 cm
175 degrees: 123.04 cm