An astronomical unit (AU) is a unit of length equal to the average distance from the Earth… — ACT

An astronomical unit (AU) is a unit of length equal to the average distance from the Earth to the Sun, approximately $1.5 \times 10^8$ kilometers. A scientific probe travels through space at a constant speed of $2.5 \times 10^4$ meters per second. How many hours would it take the probe to travel a distance of $3.6 \times 10^2$ AU?

A
$6.0 \times 10^2$
$6.0 \times 10^5$ Answer
C
$6.0 \times 10^{11}$
D
$3.6 \times 10^7$
E
$3.75 \times 10^{14}$

Answer

6.0 \times 10^5

hours

To find the travel time, we first determine the total distance in meters. Since

1\text{ AU} = 1.5 \times 10^8\text{ km}

and

1\text{ km} = 10^3\text{ m}

1\text{ AU} = 1.5 \times 10^{11}\text{ m}

. Thus, a distance of

3.6 \times 10^2\text{ AU}

equals

(3.6 \times 10^2) \times (1.5 \times 10^{11}) = 5.4 \times 10^{13}\text{ m}

. Dividing this distance by the speed of

2.5 \times 10^4\text{ m/s}

yields the travel time in seconds:

\frac{5.4 \times 10^{13}}{2.5 \times 10^4} = 2.16 \times 10^9\text{ s}

. Finally, converting seconds to hours by dividing by

3.6 \times 10^3\text{ s/h}

gives

\frac{2.16 \times 10^9}{3.6 \times 10^3} = 0.6 \times 10^6 = 6.0 \times 10^5\text{ hours}

Step-by-Step Solution

Convert the total distance from astronomical units (AU) to kilometers, and then to meters.

Distance =

3.6 \times 10^2 \text{ AU} \times 1.5 \times 10^8 \text{ km/AU} = 5.4 \times 10^{10} \text{ km} = 5.4 \times 10^{13} \text{ m}

Since the speed is given in meters per second, the total distance must be in meters to keep units consistent.

Calculate the travel time in seconds by dividing the distance in meters by the speed in meters per second.

Time in seconds =

\frac{5.4 \times 10^{13} \text{ m}}{2.5 \times 10^4 \text{ m/s}} = 2.16 \times 10^9 \text{ s}

Time is equal to distance divided by speed (

t = \frac{d}{s}

Convert the travel time from seconds to hours by dividing by

3.6 \times 10^3

(the number of seconds in one hour).

Time in hours =

\frac{2.16 \times 10^9}{3.6 \times 10^3} = 6.0 \times 10^5 \text{ hours}

Dividing the total seconds by

3,600

yields the time in hours.

Key Concept

Operations with scientific notation and dimensional analysis involving distance, rate, and time.

Alternative Method

Convert the speed to astronomical units per hour first, then divide the distance in astronomical units by this speed.

Estimated Time:2m 0s

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