Question

two cities, a and b, are 7000 miles apart. the radius of the earth is approximately 4000 miles. express angle θ in radians and in degrees. express angle θ in radians. θ = radians (type an integer or a decimal.) express angle θ in degrees. θ = ° (round to the nearest hundredth as needed.)

Explanation:

Step1: Recall arc - length formula

The arc - length formula is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians. Here, $s = 7000$ miles and $r = 4000$ miles.

Step2: Solve for $\theta$ in radians

We can rewrite the formula as $\theta=\frac{s}{r}$. Substituting $s = 7000$ and $r = 4000$, we get $\theta=\frac{7000}{4000}=\frac{7}{4}=1.75$ radians.

Step3: Convert radians to degrees

The conversion formula from radians to degrees is $\theta_{degrees}=\theta_{radians}\times\frac{180^{\circ}}{\pi}$. Substituting $\theta_{radians}=1.75$, we have $\theta_{degrees}=1.75\times\frac{180^{\circ}}{\pi}\approx1.75\times\frac{180}{3.14159}\approx100.27^{\circ}$.

Answer:

$\theta = 1.75$ radians
$\theta\approx100.27^{\circ}$