4. The wheels on Devon’s bike have r-inch radii. After the front wheel picks up a tack, Devon rolls another d feet and stops. How far above the ground is the tack?

Then,(Continuation) How many degrees does the wheel turn for each foot that it rolls?

the angle he rolls is r/2PIr (in radians)

is 180/pi * r/2PIr in degrees

How high? What is r(1- cos theta?)

Well, if the wheels on Devon's bike have r-inch radii, then we can say that the circumference of the wheel is 2πr inches.

Now, if the front wheel picks up a tack and Devon rolls another d feet, we need to figure out how far above the ground the tack is. To do that, we can use some good, old-fashioned math.

The distance that the bike travels in d feet is d times the circumference of the wheel. Since there are 12 inches in a foot, we can convert the distance to inches by multiplying d by 12.

So, the total distance traveled by the bike in inches is d times 12 times 2πr.

Now, to find how far above the ground the tack is, we need to subtract the radius of the wheel from the total distance traveled. That's because the tack would be at the center of the wheel when it picked it up.

So, the height of the tack above the ground would be (d times 12 times 2πr) - r inches.

Now, let's move on to the second part of the question - how many degrees the wheel turns for each foot it rolls.

Since we know the circumference of the wheel is 2πr inches, and there are 360 degrees in a circle, we can calculate the number of degrees the wheel turns for each inch it rolls.

The wheel would turn 360 degrees for each (2πr) inches it rolls. To find the number of degrees for each foot rolled, we can multiply that value by 12.

So, the wheel would turn (360 * 12) / (2πr) degrees for each foot it rolls.

I hope that explanation didn't tack-le you too much!

To determine how far above the ground the tack is, we need to calculate the vertical distance traveled by the front wheel of Devon's bike.

Using the formula for the circumference of a circle, the distance traveled by the front wheel can be calculated as follows:

Distance traveled by the front wheel = 2 * π * r (since the tire rolls around the circumference of the wheel)

Now, to calculate the vertical distance traveled by the front wheel, we need to consider the relationship between the circumference of the wheel and the distance it rolls.

Since the bike rolls another d feet after picking up the tack, the distance traveled by the front wheel in vertical direction can be calculated as follows:

Vertical distance traveled by the front wheel = (Distance traveled by the front wheel / Circumference of the wheel) * d

Now, let's calculate how many degrees the wheel turns for each foot that it rolls.

The wheel completes a full revolution (360 degrees) as it travels a distance equal to the circumference of the wheel. Therefore, the number of degrees the wheel turns for each foot that it rolls can be calculated using the following formula:

Degrees turned by the wheel per foot rolled = 360 degrees / Circumference of the wheel

Now that we have set up the formulas, you can substitute the appropriate values for "r" and "d" to calculate the required values.

To find the distance above the ground where the tack is, we can use the concept of similar triangles.

Let's assume the height of the tack above the ground is h inches.

Since the front wheel of Devon's bike picked up the tack, it means that the height of the front wheel from the ground is the same as the height of the tack. Let's call this distance x inches.

Now, we can set up a proportion using the similar triangles formed by the height of the front wheel and the height of the tack:

(r inches)/(x inches) = (r+h inches)/(x+d feet)

To solve for h, we can cross-multiply and solve the proportion:

(r inches) * (x+d feet) = (r+h inches) * (x inches)

Now, we can convert the measurements to a consistent unit. Since we want the height of the tack above the ground in feet, let's convert all the measurements to feet.

1 inch = 1/12 feet
1 foot = 12 inches

(r/12) * (x+d) = (r/12 + h) * x

Now, let's solve for h:

r(x+d) = (r + 12h) * 12x

rx + rd = 12rx + 12h + r12x

rd - 12h = 12rx - rx - r12x

rd - 12h = (12r - r - r12)x

rd - 12h = (24r - 2r12)x

12h = (rd - (24r - 2r12)x)

h = (rd - (24r - 2r12)x) / 12

Now we have the formula to find the height of the tack above the ground in terms of the given variables.

To find how many degrees the wheel turns for each foot that it rolls, we can use the formula for the circumference of a circle:

Circumference of a circle = 2πr

In this case, the circumference of the wheel is 2πr inches.

To find the angle in degrees that the wheel turns for each foot that it rolls, we need to convert the circumference to feet:

Circumference in feet = (2πr inches) / 12

Now, to find the angle in degrees, we can use the formula:

Angle in degrees = (Circumference in feet) / (Distance rolled in feet)

We can substitute the values and calculate the angle in degrees.