Answer:
AB = 32.6 in
Step-by-step explanation:
See the attached picture.
Triangle ABC is a right triangle, because this is a rectangular prism and height is perpendicular to top and bottom face.
Notice how we can use Pythagorean Theorem to find AB if we first calculate AC. To calculate AC we can use another right triangle ACD. (Perspective can be confusing, but angle ADC is right because this is a rectangular prism :)) From the angle ACD, AD is the opposite and AC is hypotenuse. Therefore we can use sine to find AC.
[tex]\sin{(\text{angle})} = \frac{\text{opposite}}{\text{hypotenuse}}[/tex]
[tex]\sin{(m\angle ACD)} = \frac{\text{AD}}{\text{AC}}[/tex]
[tex]\sin{(50^\circ)} = \frac{24}{\text{AC}}[/tex]
[tex]\sin{(50^\circ)} \cdot {\text{AC}} = \frac{24}{\text{AC}} \cdot {\text{AC}}[/tex]
[tex]\sin{(50^\circ)} \cdot {\text{AC}} = 24[/tex]
[tex]\frac{{\sin{(50^\circ)}} \cdot \text{AC}}{\sin{(50^\circ)}} = \frac{24}{\sin{(50^\circ)}}[/tex]
[tex]{\text{AC}} = \frac{24}{\sin{(50^\circ)}}[/tex]
[tex]{\text{AC}} \approx 31.32977 \text{ in}[/tex]
Now we can use pythagorean theorem in triangle ABC to find AB!
[tex]\text{hypotenuse}^2 = \text{leg}_1^2 + \text{leg}_2^2[/tex]
[tex]\text{AB}^2 = \text{BC}^2 + \text{AC}^2[/tex]
[tex]\text{AB}^2 = (9 \text{ in})^2 + (31.32977 \text{ in})^2[/tex]
[tex]\text{AB}^2 = 1062.5545 \text{ in}^2[/tex]
[tex]\sqrt{\text{AB}^2} = \pm\sqrt{1062.5545 \text{ in}^2}[/tex]
We are interested only in positive square root:
[tex]\text{AB} = 32.5968 \text{ in}[/tex]
Rounded to one decimal place:
[tex]\text{AB} = 32.6 \text{ in}[/tex]
