Three circular gears with radius A = 5 cm, B = 2 cm and C = 6 cm are shown.
If <CAB = 37o, how long would the arm need to be that connects the centre of gear A to the centre of gear C
To find the length of the arm that connects the center of gear A to the center of gear C, we can use the law of cosines.
Let's denote the length of the arm as x.
According to the law of cosines, in a triangle with sides a, b, and c, and angle θ between sides a and b, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(θ)
In our case, we have a right-angled triangle formed by the arm, the radius of gear A (5 cm), and the radius of gear C (6 cm). Therefore, the angle θ is 90 degrees.
Plugging in the given values into the equation above, we get:
x^2 = 5^2 + 6^2 - 2 * 5 * 6 * cos(90°)
cos(90°) equals 0, so the equation simplifies to:
x^2 = 5^2 + 6^2 - 0
Calculating:
x^2 = 25 + 36
x^2 = 61
Taking the square root of both sides:
x = sqrt(61)
Therefore, the length of the arm that connects the center of gear A to the center of gear C is approximately 7.81 cm.
To find the length of the arm that connects the center of gear A to the center of gear C, we can use the Law of Cosines. The Law of Cosines states that in any triangle ABC, with sides a, b, and c, and angle C opposite to side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In our case, if we consider gear A and C as the endpoints of the arm, we can treat the distance between their centers as side c of the triangle ABC. The radii of the gears A and C will be considered as sides a and b, respectively. The angle between sides a and b will be taken as the measure of <CAB.
Let's substitute the given values into the equation:
c^2 = a^2 + b^2 - 2ab*cos(C)
c^2 = (5cm)^2 + (6cm)^2 - 2(5cm)(6cm)*cos(37o)
Simplifying:
c^2 = 25cm^2 + 36cm^2 - 60cm^2*cos(37o)
c^2 = 25cm^2 + 36cm^2 - 60cm^2*(0.7986)
c^2 = 25cm^2 + 36cm^2 - 47.916cm^2
c^2 = 13.084cm^2
Taking the square root of both sides:
c ≈ √13.084 ≈ 3.61cm
Therefore, the length of the arm that connects the center of gear A to the center of gear C would be approximately 3.61 cm.