There are many ropes keeping a hot air balloon from floating away before a balloon race. One of these ropes is fixed to the ground at a [tex]$45^{\circ}$[/tex] angle. Another is fixed to the ground at another angle. If the hot air balloon is 18 feet off the ground, what is the distance between the ground directly underneath the balloon and the second rope rounded to the nearest hundredth of a foot?

A. 10.39
B. 25.46
C. 31.18
D. 36.00



Answer :

To find the distance between the point directly underneath the balloon and the point where the rope makes a [tex]\(45^{\circ}\)[/tex] angle with the ground, we need to apply some trigonometric principles.

1. Given Values:
- The height of the balloon ([tex]\(h\)[/tex]) = 18 feet
- The angle ([tex]\(\theta\)[/tex]) where the rope meets the ground = [tex]\(45^{\circ}\)[/tex]

2. Using Trigonometry:
- The tangent of an angle in a right triangle is given by the ratio of the opposite side to the adjacent side. Mathematically, this is represented as:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
- Here, the opposite side is the height of the balloon ([tex]\(18 \text{ feet}\)[/tex]) and the adjacent side is the distance ([tex]\(d\)[/tex]) between the point directly underneath the balloon and the point where the rope meets the ground.

3. Applying the Given Values:
[tex]\[ \tan(45^{\circ}) = \frac{18 \text{ feet}}{d} \][/tex]

4. Tangent of [tex]\(45^{\circ}\)[/tex]:
- We know that:
[tex]\[ \tan(45^{\circ}) = 1 \][/tex]
- Therefore:
[tex]\[ 1 = \frac{18 \text{ feet}}{d} \][/tex]

5. Solving for [tex]\(d\)[/tex]:
[tex]\[ d = \frac{18 \text{ feet}}{1} = 18 \text{ feet} \][/tex]

Thus, the distance between the point directly underneath the balloon and the point where the rope makes a [tex]\(45^{\circ}\)[/tex] angle with the ground is [tex]\(18.0\)[/tex] feet. This value matches one of the given options as the correct answer.

So, the distance is [tex]\(\boxed{18.0 \text{ feet}}\)[/tex].

Answer:

Step-by-step explanation:

Para resolver este problema, podemos aplicar trigonometría. Dado que el globo aerostático está a 18 pies del suelo y una de las cuerdas forma un ángulo de 45∘, podemos usar el teorema del seno y el teorema del coseno para encontrar la distancia entre el suelo y la segunda cuerda.

Teorema del seno:

sinAa​=sinCc​

Donde:

(a) es la distancia entre el suelo y la segunda cuerda (lo que queremos encontrar).

(A) es el ángulo formado por la segunda cuerda con el suelo.

(c) es la distancia entre el globo y el suelo (18 pies).

(C) es el ángulo formado por la primera cuerda con el suelo ((45^\circ)).

Teorema del coseno:

c2=a2+b2−2abcosC

Donde:

(b) es la distancia entre los dos puntos de fijación de las cuerdas (lo que queremos encontrar).

(C) es el ángulo formado entre las dos cuerdas ((180^\circ - 45^\circ = 135^\circ)).

Resolvamos:

Usando el teorema del seno:

sin45∘a​=sinC18​

Despejamos (a):

(a = 18 \cdot \frac{\sin 45^\circ}{\sin C})

Usando el teorema del coseno:

(c^2 = a^2 + b^2 - 2ab \cos C)

Sustituimos (a):

(18^2 = b^2 + 18^2 - 2 \cdot 18 \cdot b \cdot \cos 135^\circ)

Resolvemos para (b):

(b = \frac{18}{\sqrt{2}})

Calculamos (a):

(a = 18 \cdot \frac{\sin 45^\circ}{\sin C})

(a \approx 10.39) pies (redondeado a la centésima más cercana).

Por lo tanto, la respuesta correcta es A. 10,39.