Final answer:
The vertical height of a ladder can be calculated using the sine function of the angle at which it is positioned against the wall. Given the vertical height of a 15-foot ladder at a 75° angle, one can use the same sine value to determine the vertical height of a 25-foot ladder at the same angle.
Step-by-step explanation:
The question concerns the application of trigonometry to calculate the vertical height of a ladder leaning against a wall at a given angle. We're given that a 15-foot ladder set at a 75° angle against a straight wall reaches a vertical height of 14.5 feet. To find the vertical height of a 25-foot ladder set at the same angle, we can use the sine function, since the vertical height is the opposite side in relation to the angle given.
The sine of 75° (θ) is equal to the opposite side (vertical height) over the hypotenuse (length of the ladder).
θ = 75°
sin(θ) = opposite/hypotenuse
sin(75°) = vertical height / 15 feet
sin(75°) * 15 feet = 14.5 feet
Now we calculate the vertical height (h) for the 25-foot ladder maintaining the same angle:
sin(75°) = h / 25 feet
h = sin(75°) * 25 feet
We can now find the height for the 25-foot ladder by calculating the product of the sine of 75° and 25 feet.