Final answer:
To determine the height of the scoreboard, we use trigonometry by finding the tangent of the angles of elevation to the bottom (7°) and top (18°) and then multiply by the distance (320 feet) to get the height of the scoreboard which rounds to approximately 64 feet.
Step-by-step explanation:
To find the height of the scoreboard, we can use trigonometric functions involving the angle of elevation. Let's define the following variables:
- h as the height of the scoreboard
- d as the distance from the fan to the scoreboard, which is 320 feet
- θ as the angle of elevation to the bottom of the scoreboard, which is 7°
- α as the angle of elevation to the top of the scoreboard, which is 18°
Using the tangent function from trigonometry:
tan(θ) = (height of bottom of scoreboard) / d
tan(α) = (height of top of scoreboard) / d
Let x be the height of the bottom of the scoreboard from the ground, so:
x = d * tan(θ)
h + x = d * tan(α)
Subtracting the first equation from the second gives us the height h of the scoreboard:
h = d * tan(α) - d * tan(θ)
h = 320 * tan(18°) - 320 * tan(7°)
Calculating the values, we get:
h ≈ 320 * 0.32492 - 320 * 0.12278
h ≈ 102.87 feet - 39.29 feet
h ≈ 63.58 feet
Thus, rounding to the nearest foot, the height of the scoreboard is approximately 64 feet.