1.) From the top of a lighthouse
52 metres high,the angles of depression of two ships due north of it are 42 and 37. How far apart are the ships ?
Well, isn't that ship-credible! Since we have the angles of depression, we can use a little trigonometry to solve this conundrum.
First, let's imagine the Lighthouse as a starting point (and its height as a hilarious extra). Now, draw a line down to each ship creating two right-angled triangles.
Next, we can goof around with some basic trigonometric ratios. The tangent of an angle is equal to the side opposite over the side adjacent. So, if we take the tangent of the 42-degree and 37-degree angles, we can find the distances from the lighthouse to the ships.
Let's call the distance to the ship with the 42-degree angle "x" and the distance to the ship with the 37-degree angle "y".
Using the tangent function, we have:
tan(42) = 52 / x
tan(37) = 52 / y
Now, all that's left is to solve this hilarious system of equations to discover the values of x and y. Then, we can find the distance between the ships by simply subtracting the two distances.
Mathematics and humor go hand in hand, so feel free to have a laugh while you calculate the answer!
To find the distance between the two ships, we can use trigonometry and the concept of angles of depression.
Let's assume that the distance between the two ships is 'x' meters.
Now, based on the given information, we can create a right-angled triangle for each ship as follows:
For the first ship:
In the triangle, the height of the lighthouse is the opposite side, and the distance between the lighthouse and the ship is the adjacent side. The angle of depression is 42 degrees.
For the second ship:
In this triangle, the height of the lighthouse is still the opposite side, and the distance between the lighthouse and the ship is still the adjacent side. The angle of depression is 37 degrees.
Now, let's use the tangent ratio of a right-angled triangle to solve for 'x':
For the first ship:
tan(42 degrees) = opposite/adjacent
tan(42 degrees) = 52/x
For the second ship:
tan(37 degrees) = opposite/adjacent
tan(37 degrees) = 52/(x + d) (where d is the distance between the two ships)
We can solve both equations simultaneously to find the value of 'x' and 'd'.
Dividing the second equation by the first equation:
[tan(37 degrees)]/[tan(42 degrees)] = 52/(x + d) / 52/x
Simplifying this equation, we get:
tan(37 degrees)/tan(42 degrees) = x/(x + d)
Now we can substitute the values of the tangent ratios and solve for 'x':
0.7536/0.9004 = x/(x + d)
Cross-multiplying, we have:
0.7536(x + d) = 0.9004x
0.7536x + 0.7536d = 0.9004x
0.7536d = 0.9004x - 0.7536x
0.7536d = 0.1468x
Simplifying further:
d = (0.1468x) / 0.7536
Now, we substitute back the value of tan(42 degrees) = 52/x into the equation above to solve for 'd':
d = (0.1468x) / (0.7536 * (52/x))
Simplifying again:
d = 0.1468 * x * x / (52 * 0.7536)
Now, using a calculator, we can find the value of 'd'.
To find the distance between the two ships, we can use trigonometry and the concept of angles of depression.
First, let's understand the situation. We have a lighthouse with a height of 52 meters, and two ships are due north of the lighthouse. The angles of depression of the ships are given as 42 and 37 degrees, respectively.
To solve this problem, we can use tangent (tan) function, which is defined as the ratio of the opposite side to the adjacent side in a right-angle triangle.
Let's consider the first ship. The angle of depression from the top of the lighthouse to the ship is 42 degrees. We can label the distance between the lighthouse and the ship as "x1".
From the top of the lighthouse, we can draw a horizontal line to the ground, forming a right-angle triangle with the vertical side representing the height of the lighthouse (52 meters) and the horizontal side representing the distance "x1" from the lighthouse to the first ship.
Using trigonometry, we can say that the tangent of the angle of depression is equal to the opposite side (52 meters) divided by the adjacent side (x1):
tan(42) = 52/x1
Now, let's consider the second ship. The angle of depression from the top of the lighthouse to the ship is 37 degrees. Similarly, we can label the distance between the lighthouse and the second ship as "x2".
Using trigonometry again, we can say that the tangent of the angle of depression is equal to the opposite side (52 meters) divided by the adjacent side (x2):
tan(37) = 52/x2
Now, we have two equations:
tan(42) = 52/x1
tan(37) = 52/x2
To find the distances x1 and x2, we can rearrange the equations:
x1 = 52/tan(42)
x2 = 52/tan(37)
Using a calculator, we can find the values of x1 and x2. Finally, to find the distance between the two ships, we can subtract the two distances:
distance between the ships = x2 - x1
By plugging in the values and performing the calculations, we can determine the exact distance between the two ships.
distance of closer ship = 57 - tan42 = 63.3 m
distance of farther ship from lighthouse = 57/tan 37 = 75.6 m
so distance between 2 ships = 75.6-63.3 = 12.3 m
The ships re 63 and 75 m away from the lighthouse ????
TOTALLY UNREALISTIC QUESTION !