how to solve one pulley is 7 cm larger in diameter than a second pulley. the larger pulley turns at 80 rpm, and the smaller pulley turns at 136 rpm. what is the diameter of each pulley
let the diameter of the small pulley be d
let the diameter of the larger pulley be d+7
belt distance covered by smaller pulley in one minute
= 136(πd)
= 136πd
by the larger pulley
= 80(π)(d+7)
so 136πd = 80π(d+7)
divide out the π
136d = 80d + 560
etc
you state the conclusion
To solve this problem, we can use the relationship between the sizes of pulleys and their rotational speeds.
Let's establish some variables:
- Let d1 be the diameter of the smaller pulley.
- Let d2 be the diameter of the larger pulley.
Using the given information, we know that the larger pulley has a diameter 7 cm larger than the smaller pulley. So, we can create an equation:
d2 = d1 + 7
Next, we can use the ratio between the pulley sizes and the rotational speeds to set up another equation. The ratio between the pulley sizes is equal to the inverse ratio between their rotational speeds. Mathematically, this can be expressed as:
d1 / d2 = N2 / N1
Where:
- N1 is the rotational speed of the smaller pulley (136 rpm).
- N2 is the rotational speed of the larger pulley (80 rpm).
We can substitute the values into the equation and simplify:
d1 / (d1 + 7) = 80 / 136
Now, we can solve this equation to find the value of d1, which represents the diameter of the smaller pulley. Once we have that value, we can determine the diameter of the larger pulley by adding 7 to d1.
To solve this equation, multiply both sides by (d1 + 7) to eliminate the denominators:
d1 = (80 / 136) * (d1 + 7)
Now, distribute and simplify:
d1 = (80 / 136) * d1 + (80 / 136) * 7
Next, move all terms involving d1 to the left side:
d1 - (80 / 136) * d1 = (80 / 136) * 7
Simplify:
(1 - 80 / 136) * d1 = (80 / 136) * 7
Now, calculate the values on both sides of the equation:
(56 / 136) * d1 = (80 / 136) * 7
Divide both sides by (56 / 136):
d1 = (80 / 136) * 7 * (136 / 56)
Finally, calculate the value of d1:
d1 = 5.15 cm
Now that we know the diameter of the smaller pulley (d1), we can determine the diameter of the larger pulley by adding 7 cm:
d2 = d1 + 7
d2 = 5.15 cm + 7 cm
d2 = 12.15 cm
Therefore, the diameter of the smaller pulley is approximately 5.15 cm, and the diameter of the larger pulley is approximately 12.15 cm.