When a wheel is rotated through an angle of 35, a point on the circumference travels through an arc length of 2.5m. When the wheel is rotated through angles of 35 rad and 35 rev, the same point travels through arc lengths of 143 m and 9.0 x 10 m, respectively. What is the radius of the wheel?

Question

When a wheel is rotated through an angle of 35, a point on the circumference travels through an arc length of 2.5m. When the wheel is rotated through angles of 35 rad and 35 rev, the same point travels through arc lengths of 143 m and 9.0 x 10 m, respectively. What is the radius of the wheel?

Answer 1

arc lenth= angleinradians*radius

143m=35*r
solve for radius r.

Answer 2

To find the radius of the wheel, we can use the formula for arc length:

Arc Length = radius × angle

Let's calculate the radius using the given information step by step.

1. When the wheel is rotated through an angle of 35 degrees (35°), the point on the circumference travels through an arc length of 2.5 meters. We can use this information to calculate the radius.

Arc Length = 2.5 m
Angle = 35°

Substituting these values into the formula:

2.5 = radius × 35°

To solve for the radius, we need to convert the angle from degrees to radians since the next set of information is given in radians.

1 radian = (180° / π) = approximately 57.3°

2. When the wheel is rotated through an angle of 35 radians (35 rad), the same point travels through an arc length of 143 meters.

Arc Length = 143 m
Angle = 35 rad

Substituting these values into the formula:

143 = radius × 35 rad

3. Now, let's solve for the radius using the second set of information:

radius = 143 m / 35 rad

Simplifying the calculation:

radius ≈ 4.086 m (rounded to three decimal places)

4. Lastly, let's check the third set of information to confirm if the radius calculated matches. When the wheel is rotated through an angle of 35 revolutions (35 rev), the point travels through an arc length of 9.0 x 10 meters.

Arc Length = 9.0 x 10 m
Angle = 35 rev

First, we need to convert revolutions to radians. Since 1 revolution = 2π radians (a full circle), we have:

Angle = 35 rev × 2π rad/rev

Substituting this value into the formula:

9.0 x 10 m = radius × (35 × 2π) rad

Evaluating the expression:

9.0 x 10 m = radius × 70π rad

To solve for the radius:

radius = (9.0 x 10 m) / (70π rad)

Simplifying the calculation:

radius ≈ 0.406 m (rounded to three decimal places)

Thus, the radius of the wheel is approximately 0.406 m.