Question

one light year (the distance light can travel in 1 year) is approximately 9.46 × 10¹⁵ meters. approximate the time to the nearest minute for light to travel from the sun to earth if that distance is approximately 1.50 × 10¹¹ meters. min

Explanation:

Step1: Recall the formula for speed

Speed \( v \) is given by \( v=\frac{d}{t} \), where \( d \) is distance and \( t \) is time. For light, we know the distance traveled in 1 year (1 light - year) is \( d_{ly}=9.46\times 10^{15}\) meters and the time for 1 light - year is \( t_{ly}=1\) year. First, we need to find the speed of light \( c \) (in meters per minute).

First, convert 1 year to minutes.
1 year has 365 days, 1 day has 24 hours, and 1 hour has 60 minutes. So, \( t_{ly}=365\times24\times60\) minutes.
\( t_{ly}=365\times1440 = 525600\) minutes.

The speed of light \( c=\frac{d_{ly}}{t_{ly}}=\frac{9.46\times 10^{15}\text{ meters}}{525600\text{ minutes}}\)

Step2: Calculate the speed of light

\( c=\frac{9.46\times 10^{15}}{5.256\times 10^{5}}\) (since \( 525600 = 5.256\times 10^{5}\))
Using the rule of exponents \( \frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}\), we have:
\( \frac{9.46}{5.256}\times10^{15 - 5}\approx1.7998\times 10^{10}\text{ meters per minute}\)

Step3: Use the speed to find the time for Sun - Earth distance

We know the distance from Sun to Earth \( d = 1.50\times 10^{11}\) meters and speed \( c\approx1.7998\times 10^{10}\) meters per minute.
Using \( t=\frac{d}{c}\), we substitute the values:
\( t=\frac{1.50\times 10^{11}}{1.7998\times 10^{10}}\)

Step4: Simplify the time

Using the rule of exponents \( \frac{a\times10^{m}}{b\times10^{n}}=\frac{a}{b}\times10^{m - n}\), we get:
\( t=\frac{1.50}{1.7998}\times10^{11-10}\approx0.833\times 10^{1}=8.33\) minutes (rounded to the nearest minute)

Answer:

\( 8 \) (or \( 8.33\) if we consider more precise, but the question says to the nearest minute, so \( 8 \) or \( 8.33\) depending on rounding. If we do the calculation more precisely:

Alternative approach:

We know that the speed of light is approximately \( c = 3\times10^{8}\) m/s.

First, find the time in seconds: \( t=\frac{d}{c}=\frac{1.50\times 10^{11}}{3\times 10^{8}}=\frac{1.50\times 10^{11}}{3\times 10^{8}} = 500\) seconds.

Then convert seconds to minutes: \( t=\frac{500}{60}\approx8.33\) minutes, which rounds to 8 minutes (or 8.33 minutes). So the answer is approximately \( 8 \) minutes (or \( 8.33\) minutes).)