Answer:
[tex]7.7[/tex] km
Step-by-step explanation:
The submarine's path from its base forms a right triangle when its final position is "connected" to the base. We know that the right triangle has legs of [tex]6.6[/tex] km and [tex]3.9[/tex] km, and we need to find the length of its hypotenuse. To do so, we can use the Pythagorean Theorem, which states that in a right triangle, [tex]a^{2}+b^{2} =c^{2}[/tex], where [tex]a[/tex] and [tex]b[/tex] are the lengths of the triangle's legs and [tex]c[/tex] is the length of the triangle's hypotenuse. In this case, we know what [tex]a[/tex] and [tex]b[/tex] are, and we need to solve for c, so after substituting the given values of [tex]a[/tex] and [tex]b[/tex] into [tex]a^{2}+b^{2} =c^{2}[/tex] to solve for c, we get:
[tex]a^{2}+b^{2} =c^{2}[/tex]
[tex]6.6^{2} +3.9^{2} =c^{2}[/tex] (Substitute [tex]a=6.6[/tex] and [tex]b=3.9[/tex] into the equation)
[tex]43.56+15.21=c^{2}[/tex] (Evaluate the squares on the LHS)
[tex]58.77=c^{2}[/tex] (Simplify the LHS)
[tex]c^{2}=58.77[/tex] (Symmetric Property of Equality)
[tex]\sqrt{c^{2} } =\sqrt{58.77}[/tex] (Take the square root of both sides of the equation)
[tex]c=7.7,c=-7.7[/tex] (Simplify)
[tex]c=-7.7[/tex] is an extraneous solution because you can't have negative distance, if that makes sense, so therefore, the submarine is approximately [tex]7.7[/tex] km away from its base. Hope this helps!
