2. Maximum mark: Sam and Markus are on holiday exploring the coast at Wollongong. Sam has climbed the 25 m high lighthouse, which stands at the top of a cliff. Markus is in a boat some distance offshore. He measures the angle of elevation to the base of the lighthouse is 6.4°, and the angle of elevation to Sam is 10.3 a Find the distance from Markus to the base of the cliff below the lighthouse. b Find the height of the cliff.
To solve this problem, we can use trigonometry. Let's call the distance from Markus to the base of the cliff "x" and the height of the cliff "h".
a) To find the distance from Markus to the base of the cliff, we can use the tangent function. The tangent of the angle of elevation to the base of the lighthouse is equal to the opposite side (h) divided by the adjacent side (x). Therefore:
tan(6.4°) = h/x
To find x, we rearrange the equation:
x = h / tan(6.4°)
b) To find the height of the cliff, we can use the tangent function again. The tangent of the angle of elevation to Sam is equal to the opposite side (h) divided by the adjacent side (x + 25) since Sam is standing on top of the lighthouse. Therefore:
tan(10.3°) = h / (x + 25)
To find h, we rearrange the equation:
h = tan(10.3°) * (x + 25)
Now we can substitute the value of x from equation a) into equation b) to find the height of the cliff.
To find the distance from Markus to the base of the cliff below the lighthouse (known as distance AB), we can use trigonometry. Let's assume distance AB is "x."
Based on the given information, we have:
Angle of elevation to the base of the lighthouse = 6.4°
Angle of elevation to Sam = 10.3°
Using the tangent function, we can set up the following equations:
tan(6.4°) = height of the lighthouse / x
tan(10.3°) = (height of the lighthouse + 25) / x
Now, let's solve these equations step-by-step:
Step 1: Solve for x in the first equation.
x = height of the lighthouse / tan(6.4°)
Step 2: Substitute the value of x into the second equation.
tan(10.3°) = (height of the lighthouse + 25) / (height of the lighthouse / tan(6.4°))
Step 3: Simplify the equation by multiplying both sides by (height of the lighthouse / tan(6.4°)).
tan(10.3°) * (height of the lighthouse / tan(6.4°)) = height of the lighthouse + 25
Step 4: Distribute the right side of the equation.
(tan(10.3°) * height of the lighthouse) / tan(6.4°) = height of the lighthouse + 25
Step 5: Subtract 25 from both sides of the equation.
(tan(10.3°) * height of the lighthouse) / tan(6.4°) - 25 = height of the lighthouse
Step 6: Solve for the height of the lighthouse.
height of the lighthouse = (tan(10.3°) * height of the lighthouse) / tan(6.4°) - 25
Now that we have the height of the lighthouse, we can use it to find the distance AB.
Distance AB = height of the lighthouse / tan(6.4°)
This equation will give us the distance from Markus to the base of the cliff below the lighthouse (distance AB).
To find the height of the cliff (known as height BC), we can subtract the height of the lighthouse from the total height (25m).
Height BC = 25m - height of the lighthouse
I hope this explanation helps!