2440.70 – Snow Tires Versus Ordinary Tires

On Sharon's ski trip, she drove up to the ski lodge with regular tires on her electric car. The trip odometer said she went 450 miles. She recharged the battery at the lodge's charging station. She had a set of snow tires in the trunk of her car, which she hadn't had time to put on before she left. Fortunately, the roads were clear on her way to the lodge so the driving was OK.

However, while she was skiing there was a heavy snowfall, so she changed to the snow tires for her return trip, which she drove without incident. She had reset the trip odometer at the ski lodge; it said 440 miles when she got home.

Find, to the nearest hundredth of an inch, the increase in radius of the snow tires over the regular ones, if the radius of the normal ones was exactly 15 inches.

Solution

The odometer measures the number of rotations of the wheels. The problem information tells us that the wheels went around $\dfrac{440}{450}$ as many times on the return journey as on the out-going trip. That means the circumference of the tires grew to $\dfrac{440}{450}$ times what it was before.

Now $\dfrac{440}{450} = 1.0227$ . So the old circumference (call it $c$ ) times 1.0227 gives the new circumference (call it $C$ ): $C = 1.0227c$ . But also $c = 15 \cdot 2 \pi$ . If $R$ is the new radius, we find that

$\begin{align*}R &= \dfrac{C}{2 \pi} \\[1em]&= \dfrac{15 \cdot 2 \pi \cdot 1.0227}{2 \pi} \\[1em]&= 15 \cdot 1.0227 \\[0.5em]&= 15.3405\end{align*}$

So the difference in radius is about 0.34 inches (about a third of an inch).