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 label the length of the arm as x.
According to the law of cosines, we have:
x^2 = A^2 + C^2 - 2AC*cos(<CAB)
where A = 5 cm, C = 6 cm, and <CAB = 37 degrees.
x^2 = (5 cm)^2 + (6 cm)^2 - 2(5 cm)(6 cm)*cos(37 degrees)
x^2 = 25 cm^2 + 36 cm^2 - 2(30 cm^2)*cos(37 degrees)
x^2 = 25 cm^2 + 36 cm^2 - 60 cm^2*cos(37 degrees)
x^2 = 25 cm^2 + 36 cm^2 - 60 cm^2*cos(37 degrees)
x^2 = 25 cm^2 + 36 cm^2 - 60 cm^2*cos(37 degrees)
x^2 = 25 cm^2 + 36 cm^2 - 60 cm^2*(0.7986)
x^2 = 25 cm^2 + 36 cm^2 - 47.916 cm^2
x^2 = 13.084 cm^2
Taking the square root of both sides, we have:
x = √(13.084 cm^2)
x ≈ 3.62 cm
Therefore, the length of the arm that connects the center of gear A to the center of gear C is approximately 3.62 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.
First, let's label the centers of the gears as points O1, O2, and O3, where O1 is the center of gear A, O2 is the center of gear B, and O3 is the center of gear C.
Now, let's draw a line connecting the centers of gear A and C, and label this line as OC.
Since the radius of gear A is 5 cm and the radius of gear C is 6 cm, we can label the length of OC as c = 5 + 6 = 11 cm.
The angle between OC and the line connecting the centers of gear A and B, labeled as AB, is given as ∠CAB = 37°.
Similarly, we can label the length of AB as a, and since the radius of gear B is 2 cm, we have a = 2 cm.
Now, we can apply the law of cosines to find the length of the arm OC.
The law of cosines states that for any triangle with sides of lengths a, b, and c and an opposite angle θ, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(θ)
In our case, we want to find the length of OC (c), and we know the lengths of a and b, as well as the angle θ. So, we can rearrange the equation to solve for c:
c^2 = a^2 + b^2 - 2ab * cos(θ)
c^2 = (2^2) + 11^2 - 2 * 2 * 11 * cos(37°)
c^2 = 4 + 121 - 44 * cos(37°)
c^2 = 125 - 44 * cos(37°)
c ≈ √(125 - 44 * cos(37°))
Using a calculator, we can evaluate this expression:
c ≈ √(125 - 44 * cos(37°))
c ≈ √(125 - 44 * 0.7986)
c ≈ √(125 - 35.1472)
c ≈ √89.8528
c ≈ 9.485 cm
Therefore, the length of the arm that connects the center of gear A to the center of gear C is approximately 9.485 cm.