Example Question - distance between ships
Here are examples of questions we've helped users solve.
Calculating Distance between Ships Using Trigonometry
Para resolver la pregunta en la imagen, debemos encontrar la distancia \( d \) entre los barcos A y B, utilizando trigonometría. Según la información proporcionada y el dibujo, parece que debemos aplicar la ley de los cosenos. La ley de los cosenos relaciona los lados de un triángulo con el coseno de uno de sus ángulos. La fórmula es: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] donde \( c \) es el lado opuesto al ángulo \( C \), \( a \) y \( b \) son los otros lados del triángulo, y \( \cos(C) \) es el coseno del ángulo \( C \). En nuestro caso, tenemos un triángulo con lados de \( 3 \) km y \( 1.5 \) km, y un ángulo de \( 42 \) grados entre ellos. Queremos hallar \( d \), el lado opuesto al ángulo de \( 42 \) grados. Aplicando la ley de los cosenos: \[ d^2 = 3^2 + 1.5^2 - 2(3)(1.5) \cdot \cos(42^\circ) \] Calculamos \( \cos(42^\circ) \) y sustituimos los valores: \[ d^2 = 9 + 2.25 - 2(3)(1.5) \cdot \cos(42^\circ) \] \[ d^2 = 11.25 - 9 \cdot \cos(42^\circ) \] Ahora necesitamos calcular \( \cos(42^\circ) \), que aproximadamente es \( 0.743 \), y sustituirlo en la ecuación: \[ d^2 = 11.25 - 9(0.743) \] \[ d^2 = 11.25 - 6.687 \] \[ d^2 = 4.563 \] Tomando la raíz cuadrada de ambos lados para obtener \( d \): \[ d = \sqrt{4.563} \] \[ d \approx 2.136 km \] Entonces, la distancia \( d \) entre los barcos A y B es aproximadamente \( 2.136 \) kilómetros.
Calculating Distance Between Ships Using Law of Sines
The question is based on trigonometry, and it involves using the Law of Sines to solve for the distance between two ships spotted from an airplane. Given: The angles of depression to two ships are \( 32° \) and \( 40° \), respectively. The plane is 2 miles from the ship located at point A. To solve for the distance between the two ships, you need to find the length of the side opposite the given angles within the right triangles formed by the plane's altitude and the lines of sight to the ships. Firstly, since the plane is 2 miles from the ship at point A, that means the length of the line segment from the plane to ship A is 2 miles. Using the angle of depression and the fact the angle inside the right triangle formed at ship A is complementary to the angle of depression, we can determine that the angle at ship A is \( 90° - 32° = 58° \). Now, use the Law of Sines to set up the ratio of the sides of the large triangle, where \( x \) is the distance between the two ships: \[ \frac{\sin(58°)}{2\text{ miles}} = \frac{\sin(40°)}{x} \] To solve for \( x \), rearrange the equation to isolate \( x \): \[ x = \frac{2\text{ miles} \cdot \sin(40°)}{\sin(58°)} \] Now you can calculate the value of \( x \) by substituting the sine values: \[ x = \frac{2 \cdot \sin(40°)}{\sin(58°)} \] Without the exact sine values or a calculator, I cannot provide the numerical solution, but you can simply input the sine values for \( 40° \) and \( 58° \) into the formula to calculate \( x \). After you calculate it, round your answer to the nearest tenth of a mile as per the question's instructions.
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