a motorcycle has two wheels, the first has a diameter of 65cm, while the second has one of 42 cm. what is the minimal distance traveled so the two wheels has turned a natural number of turns?
Can you help me out please ?
d1 = n pi (65)
d1 = m pi (42)
65 n = 42 m
m/n is an integer
= 65/42 = 5*13 / 2*7*3
heavens to mugatroid !!! No common factors !
multiply top and bottom by 2*3*5*7*13 = 2730
since 42 and 65 have no common factors,
LCM(42,65) = 42*65 = 2730
To find the minimal distance traveled so that both wheels have turned a natural number of turns, we need to find the least common multiple (LCM) of the distances traveled by each wheel in one complete turn.
The distance traveled by a wheel in one complete turn is equal to its circumference. The formula for the circumference of a circle is C = π * d, where C is the circumference and d is the diameter.
For the first wheel:
C1 = π * d1 = π * 65 cm
For the second wheel:
C2 = π * d2 = π * 42 cm
To find the LCM of the distances traveled by both wheels, we need to convert the circumferences to a common unit. Let's convert them to centimeters (cm).
C1 = 65π cm
C2 = 42π cm
Now, we can find the LCM of these two distances. One way to find the LCM is by dividing the product of the two distances (C1 * C2) by their greatest common divisor (GCD).
Step 1: Find the GCD of C1 and C2:
To find the GCD, we need to find the greatest common factor (GCF) of 65 and 42.
Factors of 65: 1, 5, 13, 65
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
The greatest common factor (GCF) of 65 and 42 is 1.
Step 2: Calculate the LCM using the GCD:
LCM = (C1 * C2) / GCD
LCM = (65π cm * 42π cm) / 1
LCM = 2730π² cm²
Therefore, the minimal distance traveled so that both wheels have turned a natural number of turns is 2730π² cm².