Answer:
6.6 feet
Step-by-step explanation:
We know from the diagram that the angle of elevation from the ladder to the house is 49°. Since the angle of elevation is equal to the angle of depression (they are alternate interior angles), the angle of depression is also 49° (picture 1).
Now, to find side c of the triangle formed with the horizontal and Jim (picture 1), we are using the tangent trig function:
[tex]tan(\alpha )=\frac{opposite-side}{adjacent-side}[/tex]
[tex]tan(49)=\frac{5}{c}[/tex]
[tex]c=\frac{5}{tan(49)}[/tex]
[tex]c=4.35[/tex]
To find angle C (picture 2), we are using the tangent function one more time:
[tex]tan(C)=\frac{4.35}{5}[/tex]
[tex]C=arctan(\frac{4.35}{5} )[/tex]
[tex]C=41.02[/tex]
Since angle B and angle C lie on the same straight line, they are supplementary (they add to 180°), so:
[tex]B+C=180[/tex]
[tex]B+41.02=180[/tex]
[tex]B=138.98[/tex]
Now that we have the interior angle, B, of the triangle Jim, the ladder, and the board are making, we can use the law of cosines to find the shortest length, [tex]x[/tex], of the board:
[tex]x^2=5^2+2^2-2(5)(2)cos(138.98)[/tex]
[tex]x=\sqrt{5^2+2^2-2(5)(2)cos(138.98)}[/tex]
[tex]x=6.64[/tex]
And rounded to the nearest tenth:
[tex]x=6.6[/tex]
We can conclude that the shortest length of board needed is 6.6 feet.
