The eyes of a basketball player are 6 feet above the floor. The player is at the free-throw line, which is 15 feet from the center of the basket rim. What is the angle of elevation from the player's eyes to the center of the rim if the rim is 10 feet above the floor?
The basket is 10-6 = 4 feet up and 15 feet away horizontally
right triangle, base = 15, altitude = 4
tan angle = 4/15
14.93
Well, I guess it's time for some "basket-case" math! Let's calculate the angle of elevation for our basketball player.
First, we need to find the distance between the player's eyes and the center of the rim. We have a right-angled triangle, with the floor representing the base, the distance from the player to the rim as the adjacent side, and the height of the rim as the opposite side. Using the Pythagorean theorem, we can find the hypotenuse, which is the distance between the player's eyes and the rim.
Hypotenuse^2 = Adjacent^2 + Opposite^2
So, Hypotenuse^2 = 15^2 + (10-6)^2
Hypotenuse^2 = 225 + 16
Hypotenuse^2 = 241
Hypotenuse ≈ √241
Now, we can calculate the angle of elevation, using the formula:
Angle of Elevation = arc tangent (Opposite/Adjacent)
Angle of Elevation = arc tangent ((10-6)/15)
Angle of Elevation ≈ arc tangent (4/15)
Angle of Elevation ≈ 14.04°
So, the angle of elevation from the player's eyes to the center of the rim is approximately 14.04°. And with that, our basketball player can aim for a "hoop-tastic" shot!
To find the angle of elevation from the player's eyes to the center of the rim, we can create a right triangle using the given information.
Let's label the points as follows:
- The player's eyes as point A
- The center of the rim as point B
- The floor as point C
From the given information, we know:
- The distance from point A to point B is 15 feet.
- The height of point B above point C is 10 feet.
- The height of point A above point C is 6 feet.
We need to find the angle θ, which is the angle of elevation from point A to point B.
Using the tangent function, we can calculate the value of θ:
tan(θ) = opposite/adjacent
In this case, the opposite side is the height difference between points B and C, which is 10 - 6 = 4 feet. The adjacent side is the distance between points A and B, which is 15 feet.
So, we have:
tan(θ) = 4/15
To find θ, we can take the inverse tangent (or arctan) of both sides:
θ = arctan(4/15)
Using a calculator, we find that θ is approximately 14.04 degrees.
Therefore, the angle of elevation from the player's eyes to the center of the rim is approximately 14.04 degrees.
To find the angle of elevation from the player's eyes to the center of the rim, we can use trigonometry. Specifically, we can use the tangent function, which is defined as the ratio of the opposite side (height) to the adjacent side (distance).
Step 1: Find the height of the basketball player's eyes above the rim.
The height of the basketball player's eyes above the rim can be calculated by subtracting the height of the center of the rim from the height of the player's eyes:
Height above rim = Height of eyes - Height of rim = 6 feet - 10 feet = -4 feet.
Note: The negative sign indicates that the eyes are below the rim.
Step 2: Calculate the distance between the player and the rim.
The distance between the player and the rim can be determined by subtracting the distance from the center of the rim to the free-throw line from the total distance to the player's eyes from the free-throw line:
Distance = Total distance - Distance from free-throw line = 15 feet - 0 feet = 15 feet.
Step 3: Calculate the angle of elevation using the tangent function.
The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height above the rim (-4 feet), and the adjacent side is the distance between the player and the rim (15 feet):
Tangent(angle) = Opposite side / Adjacent side
Tangent(angle) = (-4 feet) / (15 feet)
Tangent(angle) = -4/15
To find the angle itself, we can find the inverse tangent (also known as arctangent or tan^(-1)) of -4/15:
Angle = arctan(-4/15) ≈ -15.94 degrees.
Note: The negative angle indicates a downward slope or angle of depression, rather than an elevation.