Question

Please help me! I don’t really have a lot a time and I need help!

Image is below

Answer 1

1. 53°

2. 18.6 ft

3. 23.3 ft

Explanation:

The image you've provided depicts a right-angled triangle, which is a classic scenario in trigonometry. We'll approach the problem using trigonometric principles and the Pythagorean theorem.

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1. Measure of the Angle at the Base of the Tree (Angle D):

The angle can be determined using the properties of triangles. In this context, the right angle is at the base of the tree, and we have one of the acute angles given as 37 degrees. The sum of angles in any triangle is 180 degrees, so the measure of the angle at the top of the tree (angle E) can be found by subtracting the given angle and the right angle from 180 degrees.

⇒ 180° - 90° - 37° = 53°

So, angle E measures 53 degrees.

2. Distance of the Bottom of the Ladder from the Base of the Tree (Segment DF):

This requires the use of trigonometry. Specifically, we will use the tangent of the given angle because it relates the opposite side (which is the height of the tree, segment DE) to the adjacent side (which is the distance from the tree we want to find, segment DF).

Recall,

`\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}`

Where θ = 37°, opposite side = 14 ft, and our unknown side is segment DF. Plugging these values in:

`\Longrightarrow \tan(37 \textdegree) = \frac{14 \ ft}{\overline{DF}}\\\\\\\\\Longrightarrow \overline{DF} = (14 \ ft)/(\tan(37 \textdegree))\\\\\\\\\therefore \overline{DF} \approx \boxed{18.6 \ ft} \text{ (rounded to the nearest tenth d.p.)}`

So, segment DF measures 18.6 ft.

3. Length of the Ladder (Segment DE):

The length of the ladder is the hypotenuse of the right triangle DEF. We can use the Pythagorean theorem to calculate this length, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

`\boxed{\left\begin{array}{ccc}\text{\underline{Pythagorean Theorem:}}\\\\a^2+b^2=c^2\end{array}\right}`

In our case, a = 18.6 ft and b = 14 ft. Plug these values into the equation above to find the length of the ladder (hypotenuse):

⇒ (18.6 ft)² + (14 ft)² = c²

⇒ 345.96 ft² + 196 ft² = c²

⇒ 541.96 ft² = c²

⇒ c = √(541.96 ft²)

∴ c = 23.3 ft (rounded to the nearest tenth d.p.)

So, the length of the ladder measures 23.3 ft.