Question 233386
After traveling an hour at 40mph the car traveled 40 miles. And after another 30 minutes it traveled another 20 miles.
Picture (or graph) the following:<ul><li>Let's make the starting point the origin (point O), with east being to the right (along the x-axis).</li><li>After the first hour the car is at the point (40, 0). Let's call this point A.</li><li>The car then starts traveling northeast. A line in this direction would make a 45 degree angle with the x-axis.</li><li>After traveling 20 miles (30 minutes), we reach another point which we'll call B.</li><li>In the triangle formed by O, A and B:<ul><li>OA = 40</li><li>AB = 20</li><li>angle OAB = 135 degrees</li><li>OB is the distance between the starting position and ending position.</li></ul></li></ul>
Once we get this above, we should see that this is a problem made for the Law of Cosines: {{{c^2 = a^2 + b^2 - 2ab*cos(C)}}}. Using this on your problem we get:
{{{(OB)^2 = (OA)^2 + (AB)^2 - 2(OA)(AB)*cos(OAB)}}}
Substituting our numbers we get:
{{{(OB)^2 = (40)^2 + (20)^2 - 2(40)(20)*cos(135)}}}
Simplifying:
{{{(OB)^2 = 1600 + 400 - 1600*((-sqrt(2))/2)}}}
{{{(OB)^2 = 2000 + 800sqrt(2)}}}
Find the square root of each side:
{{{sqrt((OB)^2) = sqrt(2000 + 800sqrt(2))}}}
{{{OB = sqrt(2000 + 800sqrt(2))}}}
This is the exact distance. I'll leave it up to you to use your calculator to come up with a decimal approximation, rounded to the nearest hundredth.