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].
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.