Answer: 88 minutes
Step-by-step explanation:
To find the time it takes for a tire to rotate 7,700 times, we need to determine the distance a tire covers in one rotation (its circumference) and then use the truck's speed to find the time.
The circumference of a tire can be found using the formula [tex]\( C = 2\pi r \)[/tex], where [tex]\( r \)[/tex] is the radius of the tire and [tex]\( \pi \approx \frac{22}{7} \)[/tex]
Given that the radius [tex]\( r = 2 \)[/tex] feet, the circumference [tex]\( C \)[/tex] is:
[tex]\[ C = 2 \times \frac{22}{7} \times 2 \][/tex]
[tex]\[ C = \frac{88}{7} \text{ feet} \][/tex]
The circumference of each tire is approximately [tex]\( \frac{88}{7} \)[/tex] or about 12.57 feet. The total distance covered by the tire after 7,700 rotations is:
[tex]\[ 7,700 \times \frac{88}{7} = 96,800 \text{ feet} \][/tex]
Now that we have the total distance, we can calculate the time it takes for the tire to rotate 7,700 times given the truck's speed of 1,100 ft/min:
[tex]\[ \text{Time} = \frac{\text{Total distance}}{\text{Speed}} \][/tex]
[tex]\[ \text{Time} = \frac{96,800 \text{ feet}}{1,100 \text{ feet per minute}} \][/tex]
[tex]\[ \text{Time} = 88 \text{ minutes} \][/tex]
Therefore, it takes 88 minutes for a tire to rotate 7,700 times.
