Question

p35. an automobile tire is rated to last for 50,000 miles. to an order of magnitude, through how many “revolutions” (complete fotations) will it turn over its lifetime? begin by estimating tire radius.

Explanation:

Step1: Estimate tire radius

A typical car tire radius is about $r = 1\ \text{ft}$ (reasonable estimate).

Step2: Calculate tire circumference

Circumference $C = 2\pi r$
$C = 2\pi(1) \approx 6\ \text{ft}$

Step3: Convert total miles to feet

1 mile = 5280 ft, so total distance $D = 50000 \times 5280 = 2.64 \times 10^8\ \text{ft}$

Step4: Find number of revolutions

Revolutions $N = \frac{D}{C}$
$N = \frac{2.64 \times 10^8}{6} = 4.4 \times 10^7$

Step5: Round to order of magnitude

Order of magnitude is the power of 10, so $10^8$ (since $4.4 \times 10^7$ is closest to $10^8$).

Answer:

The tire will turn approximately $10^8$ revolutions (order of magnitude).