A pulley system consists of a 10 inch wheel and another whose radius is unknown. A point on the rim of the wheel of unknown radius moves 450 inches when it revolves through an angle of 225 degrees. Find the angle the 10 inch wheel turns through
When you say "a 10 inch wheel" I will assume you have a radius of 10
arc length = rØ, where Ø is in radians
225° = 5π/4 radians
450 = r(5π/4)
r = 360/π inches = appr 114.59 inches (big wheel)
A point on the smaller wheel has to move 450 inches as well.
circumference = 20π
how many rotations ??
= 450/(20π) = 7.162 rotations
= appr 2578 °
Good work, but I interpret the wheel size as the diameter. If so, same steps apply, just adjusted for the smaller radius.
Well, it seems like the wheel with the unknown radius is trying to play hide and seek with us. But don't worry, with a little bit of math and humor, we'll catch it!
Let's start by finding the circumference of the unknown wheel. We know that the point on the rim of the wheel moves 450 inches. Now, a full revolution is 360 degrees, so if the point moves 450 inches when it revolves through an angle of 225 degrees, that means it has traveled half the circumference of the wheel.
So, let's set up an equation:
Circumference of the unknown wheel = 450 inches * 2
But wait, we don't actually know the radius of the unknown wheel! Uh-oh! Looks like we need to think of a clever way to find it.
Now, since we know the radius of the 10-inch wheel, let's call it R1, and let's call the radius of the unknown wheel R2. The circumference of the 10-inch wheel is given by 2 * pi * R1.
Since the point on the rim of the 10-inch wheel moves the same distance as the unknown wheel, we can set up another equation:
2 * pi * R1 = 450 inches * 2
But we want to find the angle the 10-inch wheel turns through, not its circumference!
Let's use our math superpowers to convert the distance the point on the 10-inch wheel travels into an angle. We can use the formula:
Angle = Distance / Circumference
So, the angle the 10-inch wheel turns through can be calculated as:
Angle = 2 * pi * R1 / (2 * pi * 10 inches)
Amazingly, the pi's cancel out, and we're left with:
Angle = R1 / 10 inches
Isn't math just hilarious sometimes? Now we have an expression to calculate the angle!
But wait, there's more! Remember that equation we had earlier? 2 * pi * R1 = 450 inches * 2
Well, we can substitute R1 = 10 inches into that equation to find the circumference of the 10-inch wheel. And then we can plug it into our angle formula to get the answer!
I'm going to leave the calculations to you, my friend. Have fun solving the mystery of the vanishing wheel radius and finding the angle the 10-inch wheel turns through!
To find the angle the 10-inch wheel turns through, we need to use the concept of similar triangles.
Let's denote the radius of the unknown wheel as "r."
Given that a point on the rim of the unknown wheel moves 450 inches when it revolves through an angle of 225 degrees, we can set up a proportion:
(450 inches)/(225 degrees) = (Circumference of the unknown wheel)/(Angle the unknown wheel turns through)
Now, let's calculate the circumference of the unknown wheel:
Circumference = 2πr
Substituting this into the proportion:
(450 inches)/(225 degrees) = (2πr)/(Angle the unknown wheel turns through)
To calculate the angle the unknown wheel turns through, we need to find the ratio between the circumference of the 10-inch wheel and the circumference of the unknown wheel:
10 inches / (2πr) = (10 inches / (2π)) / r
Simplifying further:
10 inches / (2πr) = 5 inches / (πr)
Now, we have an equation:
(450 inches)/(225 degrees) = (5 inches / (πr)) / (Angle the unknown wheel turns through)
To find the Angle the 10-inch wheel turns through, we can re-arrange the equation:
Angle the 10-inch wheel turns through = (450 inches) * (225 degrees) / [(5 inches / (πr))]
To complete the calculation, we need to know the value of "r" (the radius of the unknown wheel). Once we have that value, we can substitute it into the equation to find the angle.
To solve this problem, we need to use the concept of circumference and angle of rotation.
First, let's find the circumference of the 10-inch wheel. The circumference of a circle is given by the formula:
C = 2πr,
where C is the circumference and r is the radius. Plugging in the given radius of 10 inches, we get:
C = 2π(10) = 20π.
Now, let's find the radius of the other wheel. We know that a point on its rim moves 450 inches when it revolves through an angle of 225 degrees. We can set up the proportion:
(Circumference of unknown wheel) / (Angle of rotation in radians) = (Circumference of 10-inch wheel) / (360 degrees),
Let's convert 225 degrees to radians by multiplying by (π/180):
(Circumference of unknown wheel) / (225π/180) = (20π) / 360.
Simplifying the equation:
(Circumference of unknown wheel) / (5π/4) = (20π) / 360.
To find the circumference of the unknown wheel, multiply both sides of the equation by (5π/4):
Circumference of unknown wheel = (20π) / 360 * (5π/4).
Simplifying further:
Circumference of unknown wheel = π^2 / 4.
Since the circumference of a circle is given by 2πr, we can set up the equation:
2πr = π^2 / 4.
Simplifying the equation:
r = π / 8.
Now, we can find the angle the 10-inch wheel turns through by comparing the circumference of the unknown wheel to the 10-inch wheel:
(Circumference of unknown wheel) / (Circumference of 10-inch wheel) = (Angle of rotation of unknown wheel) / (Angle of rotation of 10-inch wheel).
Plugging in the values:
(π^2 / 4) / (20π) = (Angle of rotation of unknown wheel) / (360 degrees).
Simplifying the equation:
Angle of rotation of unknown wheel = (π^2 / 4) * (360 / 20π).
Simplifying further:
Angle of rotation of unknown wheel = 9 degrees.
Therefore, the 10-inch wheel turns through an angle of 9 degrees.