Question

if you shout into the grand canyon, your voice travels at the speed of sound (340 m/s) to the bottom of the canyon and back, and you hear an echo. how deep is the grand canyon at a spot where you can hear your echo 5.2 seconds after you shout? jupiter is, on average, 778,000,000 km from the sun. if the speed of light is 300,000 km/s, how long in minutes does it take light from the sun to reach jupiter?

Explanation:

Step1: Analyze sound - travel for canyon

The sound travels to the bottom of the canyon and back. The time given is for the round - trip. Let the depth of the canyon be $d$. The speed of sound $v = 340$ m/s and the total time $t=5.2$ s. The distance traveled by sound $s = v\times t$. But $s = 2d$ (round - trip).
\[s=340\times5.2\]

Step2: Solve for depth $d$

Since $s = 2d$, then $d=\frac{s}{2}$. Substituting $s = 340\times5.2$ into the formula for $d$, we get $d=\frac{340\times5.2}{2}$.
\[d = 340\times2.6=884\] m

Step3: Analyze light - travel for Jupiter

The distance between the Sun and Jupiter $s = 778000000$ km and the speed of light $v = 300000$ km/s. First, find the time in seconds using the formula $t=\frac{s}{v}$.
\[t=\frac{778000000}{300000}\text{ s}\]

Step4: Convert time from seconds to minutes

Since 1 minute = 60 seconds, to convert the time from seconds to minutes, we divide the time in seconds by 60. Let $T$ be the time in minutes.
\[T=\frac{778000000}{300000\times60}\]
\[T=\frac{7780}{180}\approx43.22\] minutes

Answer:

The depth of the Grand Canyon is 884 m.
It takes approximately 43.22 minutes for light from the Sun to reach Jupiter.