Answer:
1) Diagonal = 92.4 cm, Shorter side = 46.2 cm
2) 335.5 m
Step-by-step explanation:
Question 1
A diagonal of a rectangle divides the rectangle into two congruent right triangles, where the diagonal is the hypotenuse of the right triangles.
If the diagonal is inclined 30° from the longer side, which is 80 cm in length, then we have the side adjacent to the angle. To find the length of the diagonal (hypotenuse), we can use the cosine trigonometric ratio:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Cosine trigonometric ratio}}\\\\\sf \cos(\theta)=\dfrac{A}{H}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{$A$ is the side adjacent the angle.}\\\phantom{ww}\bullet\;\textsf{$H$ is the hypotenuse (the side opposite the right angle).}\end{array}}[/tex]
In this case:
- θ = 30°
- A = 80 cm
- H = diagonal
Substitute the values into the cosine ratio and solve for H:
[tex]\cos 30^{\circ}=\dfrac{80}{H}\\\\\\H=\dfrac{80}{\cos 30^{\circ}}\\\\\\H=92.376043070...\\\\\\H=92.4\; \sf cm[/tex]
Therefore, the diagonal of the rectangle is 92.4 cm (rounded to the nearest tenth).
To find the shorter side of the rectangle, we can use the tangent trigonometric ratio:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Tangent trigonometric ratio}}\\\\\sf \tan(\theta)=\dfrac{O}{A}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{$O$ is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{$A$ is the side adjacent the angle.}\end{array}}[/tex]
In this case:
- θ = 30°
- O = shorter side
- A = 80 cm
Substitute the values into the cosine ratio and solve for O:
[tex]\tan 30^{\circ}=\dfrac{O}{80}\\\\\\O=80 \tan 30^{\circ}\\\\\\O=46.188021535...\\\\\\O=46.2\; \sf cm\;(nearest\;tenth)[/tex]
Therefore, the shorter side of the rectangle is 46.2 cm (rounded to the nearest tenth).
[tex]\dotfill[/tex]
Question 2
The given scenario can be modelled as a right triangle, where the hypotenuse of the triangle represents the string of the kite (400 m).
Given that the angle between the ground and the string is 57°, we can use the sine trigonometric ratio to find the height of the kite (opposite side):
[tex]\boxed{\begin{array}{l}\underline{\textsf{Sine trigonometric ratio}}\\\\\sf \sin(\theta)=\dfrac{O}{H}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{O is the side opposite the angle.}\\\phantom{ww}\bullet\;\textsf{H is the hypotenuse (the side opposite the right angle).}\end{array}}[/tex]
In this case:
- θ = 57°
- O = height
- H = 400 m
Substitute the values into the sine ratio and solve for O:
[tex]\sin 57^{\circ}=\dfrac{O}{400}\\\\\\O=400 \sin 57^{\circ}\\\\\\O=335.468227178...\\\\\\O=335.5\; \sf m\;(nearest\;tenth)[/tex]
Therefore, the height of the kite is 335.5 m (rounded to the nearest tenth).
