Answer:
[tex]28.07^{\circ}[/tex]
Step-by-step explanation:
We can use basic trig. for a right triangle to solve this problem. In any right triangle, the tangent of an angle is equal to its opposite side divided by the hypotenuse (longest side) of the triangle.
The right triangle we're looking at involves the two dotted lines and an edge of the prism.
The bottom dotted line is a diagonal of the rectangular base of the prism. We can use the Pythagorean Theorem to find its length.
The Pythagorean Theorem states that for any right triangle, [tex]a^2+b^2=c^2[/tex], where [tex]a[/tex] and [tex]b[/tex] are the two legs of the triangle and [tex]c[/tex] is the hypotenuse.
In this case, the two legs are 9 and 12 and we are solving for [tex]c[/tex]:
[tex]9^2+12^2=c^2,\\81+144=c^2,\\225=c^2,\\c=15[/tex]
This is theta's adjacent side. Its opposite side is an edge of the prism labelled as 8 inches. Therefore, we have the following equation:
[tex]\tan \theta = \frac{8}{15}[/tex]
Take the inverse tangent of both sides:
[tex]\arctan(\tan \theta)=\arctan(\frac{8}{15})[/tex]
Simplify using [tex]\arctan(\tan\theta)=\theta, \theta \in (-90^{\circ}, 90^{\circ})[/tex]:
[tex]\theta=\arctan(\frac{8}{15}),\\\theta=28.07248694\approx \boxed{28.07^{\circ}}[/tex]
