Final answer:
To find how far the bicycle will move when the pedals are rotated through 135°, you calculate the gear ratio between the radii of the small gear and pedal wheel, multiply it by the proportional distance the pedal wheel covers for that rotation, and relate it to the circumference of the bicycle wheel. The bicycle will move approximately 36.708 inches based on this setup, although this does not match the provided options.
Step-by-step explanation:
To calculate how far a bicycle will move when the pedals are rotated through 135°, we need to consider the gear ratio between the small gear attached to the chain and the wheel attached to the pedals and then relate it to the circumference the bicycle wheel will cover based on this rotation.
Gear Ratio
The radius of the small gear is 1.23 inches, and the radius of the pedal wheel (the gear the pedals are attached to) is 5.62 inches. The gear ratio can be calculated using these radii:
Gear Ratio = Radius of pedal wheel / Radius of small gear
Gear Ratio = 5.62 in / 1.23 in
Gear Ratio ≈ 4.57
Each time the small gear completes a full rotation (360°), the pedal wheel will cover approximately 4.57 times less distance, since it is larger. Since the pedals are rotated through 135°, we multiply the gear ratio by the proportion of the rotation to a full circle:
Distance covered by pedal wheel = Gear Ratio × (135/360) × (Circumference of pedal wheel)
Distance covered by pedal wheel = 4.57 × (135/360) × (2 × π × 5.62 in)
We also know the radius of the bicycle wheel is 15.7 inches, so the circumference of the bicycle wheel, which gives the distance the bicycle moves per one full rotation of the wheel, is:
Circumference of bicycle wheel = 2 × π × 15.7 in ≈ 98.52 inches
To find how far the bicycle moves with the given rotation of the pedals, we then multiply the distance covered by the pedal wheel by the gear ratio:
Distance moved by bicycle = Distance covered by pedal wheel / Gear Ratio
Distance moved by bicycle = [4.57 × (135/360) × (2 × π × 5.62 in)] / 4.57
Distance moved by bicycle ≈ [1 × (135/360) × 98.52 in]
Distance moved by bicycle ≈ (135/360) × 98.52 in
Distance moved by bicycle ≈ 36.708 in
This calculation shows that the bicycle will move approximately 36.708 inches, which is not an option given. It seems there might be a misunderstanding with the question or the options provided.