A ship sailing parallel to shore sights a lighthouse at an angle of 12 degrees from its direction of travel. After traveling 5 miles farther, the angle is 22 degrees. At that time, how far is the ship from the lighthouse?
To solve this problem, we can use trigonometry and create a diagram to visualize the situation.
Let's denote the distance between the ship and the lighthouse as "d" (in miles).
From the information given, we know that the angle between the ship's direction of travel and the line connecting the ship to the lighthouse is 12 degrees when the ship is at its initial position.
Using trigonometry, we can set up the following equation:
tan(12 degrees) = d / x
where x is the distance between the ship and the lighthouse when the angle is 12 degrees.
Similarly, when the ship has traveled 5 miles farther, the angle between its direction of travel and the line connecting it to the lighthouse is now 22 degrees.
Using the same trigonometric relation, we can set up another equation:
tan(22 degrees) = (d + 5) / x
Now that we have two equations, we can solve for d and x simultaneously.
First, let's solve the first equation for x:
x = d / tan(12 degrees)
Next, let's substitute this expression for x in the second equation:
tan(22 degrees) = (d + 5) / (d / tan(12 degrees))
Now, we can solve this equation to find the value of d.
tan(22 degrees) = (d + 5) / (d / tan(12 degrees))
Rearranging the equation:
tan(22 degrees) * (d / tan(12 degrees)) = d + 5
Multiplying through by tan(12 degrees):
tan(22 degrees) * d = (d + 5) * tan(12 degrees)
Expanding and rearranging:
d * (tan(22 degrees) - tan(12 degrees)) = 5 * tan(12 degrees)
Finally, solving for d:
d = (5 * tan(12 degrees)) / (tan(22 degrees) - tan(12 degrees))
Using a calculator, we can evaluate this expression:
d ≈ 6.79 miles
Therefore, the ship is approximately 6.79 miles away from the lighthouse when the angle is 22 degrees.
To find the distance between the ship and the lighthouse, we can use the concept of trigonometry. Let's assume that the distance between the ship and the lighthouse is represented by 'x' miles.
From the given information, we have two angles: 12 degrees and 22 degrees. We can consider these angles as the angles of elevation from the ship to the top of the lighthouse.
Now, let's analyze the scenario after the ship has traveled 5 miles farther. At this point, we can create a right-angled triangle with the ship, the lighthouse, and the point of observation.
Using trigonometry, we can use the tangent function to relate the angle of elevation to the opposite and adjacent sides of a right triangle. The tangent function is defined as the ratio of the opposite side over the adjacent side.
For the initial angle of 12 degrees, we have the tangent of 12 degrees equal to the opposite side (x miles) divided by the ship's original position from the lighthouse.
tan(12) = x / original distance (from lighthouse)
Similarly, for the angle of 22 degrees, we have:
tan(22) = x / (original distance + 5 miles)
We can rearrange these equations to solve for x:
x = tan(12) * original distance
x = tan(22) * (original distance + 5)
Since both equations equal x, we can set them equal to each other:
tan(12) * original distance = tan(22) * (original distance + 5)
By solving this equation algebraically, we can find the original distance between the ship and the lighthouse.
Let the lighthouse be h miles from shore. Let x be the distance of the ship from a point on shore closest to the lighthouse.
h/(x+5) = tan 12°
h/x = tan 22°
so, equating h,
(x+5)tan 12° = x tan 22°
.212(x+5) = .404x
1.06 = .282x
x = 3.759 mi
Now, the distance of the ship from the lighthouse, d, can be found by
3.759/d = cos 22°
d = 4.05 mi