Devon’s bike has wheels that are 27 inches in diameter. After the front wheel picks up a tack, Devon rolls another 100 feet and stops. How far above the ground is the tack?
reiny the diameter is 27, so radius = 13.5
Well, if Devon's bike wheel picked up a tack and he rolled another 100 feet, I'd say that tack is pretty high up! Probably somewhere in the clouds, hanging out with unicorns and rainbows. It's reached new heights, quite literally! But in all seriousness, without more information, it's impossible to determine the exact height of the tack.
To find the height at which the tack is above the ground, we need to calculate the vertical distance the bike traveled when the front wheel picked up the tack.
First, let's convert the diameter of the wheel to a radius: 27 inches / 2 = 13.5 inches.
Next, let's convert the radius to feet: 13.5 inches / 12 inches/foot = 1.125 feet.
Now we need to calculate the circumference of the wheel: circumference = 2 * π * radius.
circumference = 2 * 3.14 * 1.125 feet ≈ 7.07 feet.
The circumference of the wheel represents the distance covered by one complete rotation. So, when the front wheel picks up the tack, the bike has traveled a distance of 7.07 feet.
After the front wheel picks up the tack, Devon rolls another 100 feet and stops. This means that the wheel rotated approximately 100 feet / 7.07 feet/rotation = 14.14 rotations.
Since the tack is fixed to the wheel, it will also rotate with the wheel. Therefore, the tack must have rotated 14.14 rotations.
To find the height at which the tack is above the ground, we need to multiply the number of rotations by the circumference of the wheel:
height = 14.14 rotations * 7.07 feet/rotation = approximately 100 feet.
Thus, the tack is approximately 100 feet above the ground.
To find out how far above the ground the tack is, we can use the concept of a circle's circumference.
First, let's find the circumference of the front wheel. The formula for the circumference of a circle is C = π * d, where C is the circumference and d is the diameter.
In this case, the diameter of the front wheel is 27 inches. So, the circumference of the front wheel is C = π * 27 inches.
Next, let's convert the distance Devon rolled (100 feet) to inches because we need to use the same unit of measurement. There are 12 inches in 1 foot, so 100 feet is equal to 100 * 12 = 1200 inches.
Now, if Devon rolls one complete revolution of the front wheel, the bike moves a distance equal to one circumference. Therefore, if Devon rolls 1200 inches, it is equal to 1200 / C revolutions of the wheel.
To find out the height of the tack, we need to calculate the distance traveled vertically by the circumference of the wheel.
The formula to calculate the height is: height = circumference * revolutions.
height = C * (1200 / C) = 1200 inches.
So, the tack on the front wheel is 1200 inches above the ground.
100 ft = 1200 inches
one rotation covers 27π or 84.823 inches
so the wheel went 1200/84.823 or 14.1471 rotations
We are interested in the .1471 part of the last rotation.
make a sketch of the circle, draw in vertical radius, and label an arc length of .1471(84.823) or 12.478 inches.
We need the central angle which subtends that arc of 12.478
from arc = rØ
we have
12.478 = 27Ø
Ø = .46215 radians
From the end of the arc, draw a horizontal cutting the radius to create a right-angled triangle
let h be the length form the centre of the circle to that horizontal
cos .46215 = h/27
h = 27cos.4615 = 24.16
so the nail is 27 - 24.16 or 2.8 inches above the ground.
You better check my arithmetic.