Q. 58
Question
Ian is doing a high traverse. One morning he looks at the map and notes that if he considers his camp to be at the origin, then his objective is at \((5.9, 3.3, −0.37)\). All distances are in miles.
(a) How far away is his objective, as the crow flies?
(b) In order to reach his objective, Ian has to go over a high pass that lies at \((4.2, 4.4, 0.15)\) relative to his camp. Find a more realistic estimate of how far he has to go to his objective than that from part (a).
Step-by-Step Solution
VerifiedPart (a) As the crow flies the objective is \(6.8 miles\) far away from Ian.
Part (b) The more realistic estimate of Ian has to go to his objective than that from part (a) is \(8.17 miles\).
It is given that the camp of Ian is at the origin and his objective is at \((5.9, 3.3, −0.37)\). We have to find the distance between his objective and the camp.
To find the distance between his objective and the camp. We will use the distance formula.
So, the distance between two points is
\(=\sqrt{\left ( 5.9-0 \right )^{2}+\left ( 3.3-0 \right )^{2}+\left ( -0.37-0 \right )^{2}}\)
\(=\sqrt{34.81+10.89+0.1369}\)
\(=\sqrt{45.8369}\)
\(\approx 6.8 miles\)
We have to find the distance between the camp and the objective through the given point \((4.2, 4.4, 0.15)\).
Now, let's first find the distance between the camp \((0, 0, 0)\) and the point \((4.2, 4.4, 0.15)\).
So,
\(=\sqrt{\left ( 4.2-0 \right )^{2}+\left ( 4.4-0 \right )^{2}+\left ( 0.15-0 \right )^{2}}\)
\(=\sqrt{17.64+19.36+0.0225}\)
\(=\sqrt{37.0225}\)
\(\approx 6.08\)
Now, let's first find the distance between the objective \((5.9, 3.3, −0.37)\) and the point \((4.2, 4.4, 0.15)\).
So,
\(=\sqrt{\left ( 5.9-4.2 \right )^{2}+\left ( 3.3-4.4 \right )^{2}+\left ( -0.37-0.15 \right )^{2}}\)
\(=\sqrt{2.89+1.21+0.2704}\)
\(=\sqrt{4.3704}\)
\(=2.09\)
Therefore, the distance between the camp and the objective through the given point \((4.2, 4.4, 0.15)\) is \(6.08+2.09=8.17 miles\).