Please help...a computerized spin balance machine rotates a 25 inch diameter tire at 480 revolutions per minute. (a) find the road speed in miles per hour at which the tire is being balanced (b) at what rate should the spin balance machine be set so that the tire is being tested for 70 miles per hour?

one rotation covers 25π inches

so in 1 minute it covers 12000π inches
and in 1 hour it covers 720000π inches

1 mile = 5280 ft = 63360 inches

so it goes 720000π/63360 miles/hour = 35.7 mph

since the road speed varies directly with the rmp

35.7/480 = 70/x
35.7x = 33600
x = 33600/35.7 = appr 941 rpm

Why did the tire go to the therapist? Because it needed some "wheel" balancing! Let's calculate the road speed for both scenarios:

(a) To find the road speed in miles per hour when the tire is being balanced, we need to calculate the circumference of the tire. The formula for the circumference of a circle is C = πd, where C is the circumference and d is the diameter.

So, C = π * 25 inches.

Now, let's convert the circumference from inches to miles. There are 12 inches in a foot, and 5,280 feet in a mile. Therefore, there are 12 * 5,280 = 63,360 inches in a mile.

To convert the circumference to miles, we divide it by the number of inches in a mile:
C_miles = C / 63,360.

Now we need to find the distance traveled in an hour, which is equal to the circumference times the number of revolutions per minute (rpm), multiplied by the number of minutes in an hour:
Distance = (C_miles * rpm) * 60.

Substituting the values we have, we get:
Distance = (C / 63,360 * 480) * 60.

(b) To find the rate at which the spin balance machine should be set for a road speed of 70 miles per hour, we do the same calculations as before, but with a different rpm value:
Distance = (C / 63,360 * rpm) * 60.

Now we know the formula, but it seems we're missing something. Could you please provide the rpm value for the road speed of 70 mph? I just hope it's not asking the machine to break the sound barrier!

To find the road speed in miles per hour at which the tire is being balanced, we can use the formula for the circumference of a circle:

C = πd

where C is the circumference and d is the diameter.

Given that the diameter of the tire is 25 inches, we can calculate the circumference:

C = π(25) = 78.54 inches

Now, to convert the road speed from revolutions per minute to miles per hour, we use the formula:

Speed (miles per hour) = Circumference (miles) × Revolutions per minute × 60 minutes per hour

The circumference needs to be converted from inches to miles:

Circumference (miles) = Circumference (inches) ÷ 12 inches per foot ÷ 5280 feet per mile

Substituting the values:

Circumference (miles) = 78.54 inches ÷ 12 inches per foot ÷ 5280 feet per mile = 0.001244 miles

Now, we can find the road speed in miles per hour:

Speed (miles per hour) = 0.001244 miles × 480 revolutions per minute × 60 minutes per hour
Speed (miles per hour) = 0.001244 × 480 × 60 ≈ 35.5 miles per hour

Therefore, the road speed at which the tire is being balanced is approximately 35.5 miles per hour.

Now, to find the rate at which the spin balance machine should be set so that the tire is being tested for 70 miles per hour, we rearrange the formula:

Speed (miles per hour) = Circumference (miles) × Revolutions per minute × 60 minutes per hour

Rearranging for revolutions per minute:

Revolutions per minute = Speed (miles per hour) ÷ (Circumference (miles) × 60 minutes per hour)

Substituting the given values:

Revolutions per minute = 70 miles per hour ÷ (0.001244 miles × 60) ≈ 94037 revolutions per minute

Therefore, the spin balance machine should be set to approximately 94037 revolutions per minute to test the tire for a speed of 70 miles per hour.

To find the road speed in miles per hour at which the tire is being balanced, we can use the formula:

Road Speed = π * Diameter * RPM * 1 hour / 63360 inches

Let's calculate it step by step.

(a) Road Speed at which the tire is being balanced:
Diameter of the tire = 25 inches
RPM (Revolutions per minute) = 480

First, let's convert the diameter from inches to feet:
Diameter (in feet) = 25 inches / 12 inches per foot = 2.08 feet

Now, let's calculate the road speed using the formula:
Road Speed = π * 2.08 feet * 480 RPM * 1 hour / 63360 inches

We know that π (pi) is approximately 3.14159.
Road Speed ≈ 3.14159 * 2.08 feet * 480 RPM * 1 hour / 63360 inches
Road Speed ≈ 26.18154 feet per hour / 63360 inches

To convert the result to miles per hour, we need to divide by the number of feet in a mile:
Road Speed ≈ 26.18154 / 5280 miles per hour
Road Speed ≈ 0.0049519 miles per hour

Therefore, the road speed at which the tire is being balanced is approximately 0.0049519 miles per hour.

(b) To find the rate at which the spin balance machine should be set so that the tire is being tested for 70 miles per hour, we need to rearrange the formula:

Machine Rate = (Road Speed * 63360 inches) / (π * Diameter * 1 hour)

Let's calculate it step by step.

Road Speed (desired) = 70 miles per hour
Diameter of the tire = 25 inches

First, let's convert the diameter from inches to feet:
Diameter (in feet) = 25 inches / 12 inches per foot = 2.08 feet

Now, let's convert the road speed from miles per hour to inches per hour:
Road Speed (desired) = 70 miles per hour * 5280 feet per mile * 12 inches per foot
Road Speed (desired) = 44,9280 inches per hour

Now, let's calculate the machine rate using the formula:
Machine Rate = (44,9280 inches per hour * 63360 inches) / (π * 2.08 feet * 1 hour)

We know that π (pi) is approximately 3.14159.
Machine Rate ≈ (44,9280 * 63360) / (3.14159 * 2.08)
Machine Rate ≈ 1441668480 / 6.50545832
Machine Rate ≈ 221,544,150.59 RPM

Therefore, the rate at which the spin balance machine should be set so that the tire is being tested for 70 miles per hour is approximately 221,544,151 RPM.