For this problem, we are given a rectangle with the measurement of its dyagonal and the angle between the dyagonal and the base. We need to determine the perimeter and area for this rectangle.
For this, we need to analyze the right triangle that is formed between the dyagonal, the width and height of the rectangle. This triangle is shown below:
From the image above, we can notice that the height is the opposite side to the known angle. Therefore we can calculate it by using the sine relation on a right triangle.
[tex]\begin{gathered} \sin 29=\frac{\text{ height}}{18} \\ \text{height}=18\cdot\sin 29 \\ \text{height}=18\cdot0.48 \\ \text{height}=8.73 \end{gathered}[/tex]
The rectangle's height is equal to 8.73 ft.
On the other hand the width is the adjascent side to the known angle, therefore we can calculate it by using the cossine relation.
[tex]\begin{gathered} \cos 29=\frac{\text{ width}}{18} \\ \text{width}=18\cdot\cos 29 \\ \text{width}=18\cdot0.87 \\ \text{width}=15.74 \end{gathered}[/tex]
The rectangle's width is equal to 15.74 ft.
Now we can calculate the perimeter and area for the rectangle.
[tex]\begin{gathered} P=2\cdot(\text{width}+\text{height)} \\ P=2\cdot(15.74+8.73) \\ P=2\cdot24.47 \\ P=48.94\text{ ft} \end{gathered}[/tex]
The perimeter for the rectangle is 48.94 ft.
[tex]\begin{gathered} A=\text{width}\cdot\text{height} \\ A=15.74\cdot8.73 \\ A=137.41 \end{gathered}[/tex]
The area for the rectangle is 137.41 square ft.
