From a boat on the river below a dam, the angle of elevation to the top of the dam is 15 degrees 35'. if the dam is 2083 feet above the level of the river how far is the boat from the base of the dam to the nearest foot
how about
tan 15°35' = 2083/x, where x is the horizontal distance you want
x = 2083/tan
= 2083/.278891.... = 7468.85 ft or 7469 ft to the nearest foot
check:
tan^-1 (2083/7468.85) = 15.58333...°
= 15° .58333...*60'
= 15° 35'
tan 15 35/60 deg = 2083 / d
so
d = 2083 / 0.27889
To find the distance from the boat to the base of the dam, we can use trigonometry. In this case, we will use the tangent function.
First, let's convert the angle of elevation to decimal form:
15 degrees 35' = 15 + (35/60) = 15.583 degrees
Next, let's set up the trigonometric equation:
tan(angle) = opposite / adjacent
In this scenario, the opposite side is the height of the dam (2083 feet), and the adjacent side is the distance from the boat to the base of the dam.
Let's denote the distance from the boat to the base of the dam as 'x'. The equation becomes:
tan(15.583 degrees) = 2083 / x
Now, we need to solve for 'x':
x = 2083 / tan(15.583 degrees)
Using a scientific calculator or mathematical software, we can evaluate the tangent of 15.583 degrees:
tan(15.583 degrees) ≈ 0.2733
Therefore:
x ≈ 2083 / 0.2733 ≈ 7620
Therefore, the boat is approximately 7620 feet away from the base of the dam, rounded to the nearest foot.