A boat is 300m away from the foot of a cliff. The angle of elevation from the boat to the top of the cliff is 16 degrees.
a) Determine the height of the cliff to the nearest meter.
b) If the boat sails 75 m closer to the cliff, determine , to the nearest degree, the new angle of elevation of the cliff top from the boat.
a) h / 300 = tan(16º)
... h = 300 * tan(16º)
b) tan(Θ) = h / (300 - 75)
To solve this problem, we can use trigonometry. Let's start with the first question:
a) To determine the height of the cliff, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the cliff, and the adjacent side is the distance between the boat and the foot of the cliff.
We can set up the equation as follows:
tan(16 degrees) = height / 300m
To find the height, we can rearrange the equation to solve for it:
height = tan(16 degrees) * 300m
Using a calculator, we can evaluate this expression:
height = (0.2867) * 300m
height = 86m
Therefore, the height of the cliff is approximately 86 meters.
b) To determine the new angle of elevation, we can use the inverse tangent function. Given that the boat has moved 75 meters closer to the cliff, the new distance between the boat and the foot of the cliff is 300m - 75m = 225m.
We can set up the equation as follows:
tan(new angle) = height / 225m
To find the new angle, we can use the inverse tangent function on both sides:
new angle = arctan(height / 225m)
Using a calculator, we can evaluate this expression:
new angle = arctan(86m / 225m)
new angle = 21.64 degrees
Therefore, the new angle of elevation of the cliff top from the boat is approximately 22 degrees.
To solve this problem, we can use trigonometry, specifically the tangent function. The tangent of an acute angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
Let's break down the problem step by step:
a) Determine the height of the cliff to the nearest meter.
Step 1: Calculate the length of the side opposite the 16-degree angle.
We know that the angle of elevation from the boat to the top of the cliff is 16 degrees. Let's call the height of the cliff h. Therefore, h is the side opposite the 16-degree angle.
Step 2: Find the length of the side adjacent to the 16-degree angle.
The distance between the boat and the foot of the cliff is given as 300 meters. Let's call this side x. Therefore, x is the side adjacent to the 16-degree angle.
Step 3: Use the tangent function to find the height of the cliff.
The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, we can write:
tan(16 degrees) = h / x
Now, we can rearrange the equation to solve for h:
h = x * tan(16 degrees)
Plugging in the values, we have:
h = 300 * tan(16 degrees)
Using a scientific calculator or an online trigonometry calculator, we find that tan(16 degrees) ≈ 0.3064.
So, h ≈ 300 * 0.3064 ≈ 91.93 meters.
Therefore, the height of the cliff to the nearest meter is approximately 92 meters.
b) If the boat sails 75 m closer to the cliff, determine the new angle of elevation of the cliff top from the boat.
When the boat sails 75 meters closer to the cliff, the distance between the boat and the foot of the cliff becomes 300 - 75 = 225 meters. Let's call this new distance y.
Now we can repeat the process for finding the angle of elevation, but with y as the adjacent side:
tan(new angle) = h / y
Given that we found the height of the cliff to be approximately 92 meters, and the new distance y is 225 meters, we can solve for the new angle of elevation:
tan(new angle) = 92 / 225
Using a calculator, we find that the inverse tangent of 92/225 is approximately 21.62 degrees.
Therefore, to the nearest degree, the new angle of elevation of the cliff top from the boat is approximately 22 degrees.