From a boat on the lake, the angle of elevation to the top of a cliff is 24 degree 19'. If the base of the cliff is 2994 feet from the boat, how high is the cliff (to the nearest foot)?
Draw a diagram
Review your basic trig functions. You can see that
h/2994 = tan 24°19'
h = 1352.9 ft
To solve this problem, we can use trigonometry, specifically the tangent function. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
In this case, the angle of elevation is given as 24 degrees 19'. To convert this to decimal degrees, we multiply the number of degrees by 60, then add the number of minutes. So, 24 degrees 19' is equal to 24 + (19/60) = 24.317 degrees.
Let's assume that h is the height of the cliff. We can set up a right triangle with the cliff as the perpendicular side, the distance from the boat to the base of the cliff as the adjacent side, and the height of the cliff as the opposite side. Since we want to find the height of the cliff, we'll solve for h.
Using trigonometry, the tangent of the angle of elevation is equal to the height of the cliff divided by the distance from the boat to the base of the cliff:
tan(24.317 degrees) = h / 2994 feet
To solve for h, we can rearrange the equation:
h = tan(24.317 degrees) * 2994 feet
Using a scientific calculator, we can find the tangent of 24.317 degrees:
tan(24.317 degrees) ≈ 0.4543
Now, we can plug in this value and the given distance into the equation to find the height of the cliff:
h = 0.4543 * 2994 feet
h ≈ 1,361.52 feet
Therefore, the height of the cliff, to the nearest foot, is approximately 1,362 feet.