The circumference of a rear wheel of a carriage is 50 inches more than a front wheel. If the rear wheel makes as many revolutions in going 200 feet as the front wheel in going 120 feet, find the length of the circumference of each wheel.
Radius of large wheel --- R
radius of smaller wheel -- r
Circumf of large = 2πR
circumf of small = 2πr
2πR - 2πr = 50
R - r = 25/π
R = 25/π + r
rotations of front wheel = 1440/(2πr) = 720/(πr)
rotations of rear wheel = 2400/(2πR) = 1200/(πR)
but 1200/πR = 720/πr
1200πr = 720πR
5r = 3R
3(25/π + r) =5r
75/π + 3r = 5r
r = 75/2π = appr 11.9366
R = appr 19.8944
carry on
To find the circumference of each wheel, we can set up a system of equations based on the given information.
Let's assume the circumference of the front wheel is x inches. Since the circumference of the rear wheel is 50 inches more than the front wheel, its circumference would be (x + 50) inches.
Now, we know that the rear wheel makes the same number of revolutions as the front wheel. In other words, the distance covered by each wheel in one revolution will be the same.
The distance covered by the front wheel in one revolution is equal to its circumference (x inches). Similarly, the distance covered by the rear wheel in one revolution is equal to its circumference ((x + 50) inches).
Using the above information, we can set up the following equation:
x inches (front wheel circumference) * n (number of revolutions) = 120 feet
(x + 50) inches (rear wheel circumference) * n (number of revolutions) = 200 feet
Now, let's convert the given distances from feet to inches to make the units consistent:
120 feet = 120 * 12 inches = 1440 inches
200 feet = 200 * 12 inches = 2400 inches
The equations become:
x * n = 1440
(x + 50) * n = 2400.
We have a system of two equations with two unknowns (x and n). We can now solve this system of equations to find the values of x (circumference of the front wheel) and (x + 50) (circumference of the rear wheel).
Dividing the second equation by the first equation, we get:
(x + 50) / x = 2400 / 1440
(x + 50) / x = 5/3.
Cross-multiplying, we have:
3(x + 50) = 5x
3x + 150 = 5x
150 = 5x - 3x
150 = 2x.
Dividing both sides of the equation by 2:
150 / 2 = x
75 = x.
Therefore, the circumference of the front wheel (x) is 75 inches. The circumference of the rear wheel would be (75 + 50) inches = 125 inches.
Hence, the length of the circumference of each wheel is 75 inches for the front wheel and 125 inches for the rear wheel.