these two are the same type of questions but i have no clue how to figure out the answer. i have tried drawing it out but don't know what to do after that.
1. a bicycle with a 28-inch wheel(diameter) travels a distance of 800 feet. how many revolutions does the wheel make (to the nearest revolution)?
2. A wheel with a 20-inch radius is marked at two points on the rim. the distance between the marks along the wheel is found to be 3 inches. what is the angle (to the nearest tenth of a degree) between the radii to the two marks?
it would be realy helpful if you could explain to me how to figure out these two. thks
1. How far does the wheel go in 1 rotation?
C = piD = pi(28)inches
but 800 feet = 9600 inches
so no of rotations = 9600/(28pi) = 109 to the nearest whole number
for 2. is the 3 inches the straight line distance betweeen the points or is it measured along the arc?
If straight, use the Cosine Law,
If arc length, set up a ratio..
angle/360 = 3/(40pi), solve for "angle"
Sure, I can help you with both questions. Let's break them down step by step.
1. Finding the number of wheel revolutions:
To find the number of revolutions, we need to calculate the circumference of the wheel and divide the total distance traveled by the circumference. The formula for the circumference of a circle is C = πd, where C is the circumference and d is the diameter.
Given that the diameter of the wheel is 28 inches, we can calculate the circumference:
C = π * d = π * 28 inches.
Now, we need to convert the distance traveled from feet to inches since the circumference is in inches. There are 12 inches in a foot, so multiply the distance by 12:
Distance traveled = 800 feet * 12 inches/foot = 9,600 inches.
Now, to find the number of revolutions, divide the distance traveled by the circumference of the wheel:
Number of revolutions = Distance traveled / Circumference of wheel.
Substituting the values, we get:
Number of revolutions = 9,600 inches / (π * 28 inches).
To get the answer to the nearest revolution, calculate this value. Remember to use the approximate value of π as 3.14 or 22/7 to simplify the calculations.
2. Finding the angle between the radii:
To find the angle between the radii, we need to use the formula for the central angle of a circle. The central angle is directly proportional to the arc length (in this case, the distance between the marks on the wheel). The formula is Angle = (Arc Length / Circumference) * 360 degrees.
Given that the radius of the wheel is 20 inches and the distance between marks on the wheel is 3 inches, we can calculate the angle:
Angle = (3 inches / Circumference of wheel) * 360 degrees.
Again, we need to calculate the circumference of the wheel using the formula C = πd, substituting the value of the diameter:
C = π * (2 * radius) = π * (2 * 20 inches).
Now, substitute this value back into the formula for the angle:
Angle = (3 inches / (π * 40 inches)) * 360 degrees.
Simplify and solve this equation to find the angle. Remember to use the approximate value of π as 3.14 or 22/7 for accurate results.
Following these steps, you'll be able to find the solutions to both questions.