Question

Please help with these 2 questions-

Answer 1

1) h = 7.8 yards

2) t = 23.8 feet

Explanation:

Question 1

In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the side opposite that angle to the length of the hypotenuse.

`\boxed{\begin{array}{l}\underline{\textsf{Sine trigonometric ratio}}\\\\\sf \sin(\theta)=(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}}`

In the given right triangle:

• Angle = 15°
• Side opposite the angle = h
• Hypotenuse = 30 yards

Therefore, to find the vertical distance (h), we can use the sine ratio:

`\sin 15^(\circ)=(h)/(30)`

Solve for h:

`30\sin 15^(\circ)=h`

`h=30\sin 15^(\circ)`

`h=7.764571353...`

`h=7.8\; \sf yards\;(nearest\;tenth)`

So, the vertical distance (h) is 7.8 yards (rounded to the nearest tenth).

`\hrulefill`

Question 2

In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the side opposite that angle to the length of the side adjacent that angle.

`\boxed{\begin{array}{l}\underline{\textsf{Tangent trigonometric ratio}}\\\\\sf \tan(\theta)=(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}}`

In the given right triangle:

• Angle = 50°
• Side opposite the angle = t
• Side adajcent the angle = 20 feet

Therefore, to find the height of the tree (t), we can use the tangent ratio:

`\tan 50^(\circ)=(t)/(20)`

Solve for t:

`20 \tan 50^(\circ)=t`

`t=20 \tan 50^(\circ)`

`t=23.83507185...`

`t=23.8\; \sf feet\;(nearest\;tenth)`

So, the height of the tree (t) is 23.8 feet (rounded to the nearest tenth).