A 5 foot tall woman stands 15ft from a statue. She must look up at an angle of 60 degrees to see the top of the statue. How tall is the statue?
To find the height of the statue, we can use the concept of similar triangles. Let's denote the height of the statue as "h".
In the given scenario, we have two similar right triangles:
1. The first triangle consists of the woman's line of sight, the distance between the woman and the statue, and the height of the woman.
We know that the height of the woman is 5 ft, and she stands 15 ft away from the statue.
2. The second triangle consists of the woman's line of sight, the distance between the woman and the statue, and the height of the statue (h).
We can set up the following proportion based on the concept of similar triangles:
(height of the statue) / (distance between the woman and the statue) = (height of the woman) / (distance between the woman and the statue)
Mathematically, this can be represented as:
h / 15 ft = 5 ft / 15 ft
We can simplify this equation to:
h / 15 ft = 1/3
To solve for the height of the statue (h), we can cross multiply:
h = 15 ft * (1/3)
h = 5 ft
Therefore, the height of the statue is 5 ft.
To find the height of the statue, we can use trigonometry. Let's call the height of the statue "h".
First, let's break down the problem. We have a right triangle with the height of the woman (5 feet), the distance from the woman to the statue (15 feet), and the angle at which she must look up (60 degrees).
We can use the tangent function since we have the opposite side (height of the woman) and the adjacent side (distance to the statue).
Using the tangent function:
tan(60 degrees) = opposite / adjacent
tan(60 degrees) = 5 / 15
Next, we can solve the equation for the ratio of the opposite side to the adjacent side:
tan(60 degrees) = h / 15
Finally, we can solve for "h" by multiplying both sides of the equation by 15:
h = 15 * tan(60 degrees)
Now, let's calculate the height of the statue:
h = 15 * tan(60 degrees)
h ≈ 15 * 1.73 (approximately)
h ≈ 25.95
Therefore, the height of the statue is approximately 25.95 feet.
Just set it up like a right triangle problem.
In fact this problem is a special triangle problem dealing with 30,60,90 degree triangles.
I recommend drawing a picture with these dimensions:
*Your x leg is 15ft
*y leg is the height of the statue (what you want to find), for now represent this as y
*Your angle of 60 degrees will be the one near the woman's eyes
*The 30 degree angle will be the one near the top of the statue when you draw the triangle
Now recall the special relationships with all the angles and the sides
*the side across from the 30 degree angle = x
*the side across from the 60 degree angle = x√3
In this situation you have here you can see that the side across from the 30 degree angle is 15ft so that means:
15ft = x
This means that 15√3 = x√3, the side across from the 60 degree angle so this is the part of the height of the statue
Now, I say part of the height because the woman is viewing the statue at a height of 5 feet so you need to add on that 5 feet to the statue's height as well.
You should end up with 15√3 + 5 ft, if you need the decimal approximate you can type it into your calculator to find out.
Let h be the height of the statue which lies above the woman's height of 5 ft
h/15 = tan60°
h = 15tan60
= 15√3
so the height of the statue is 15√3 + 5