Final answer:
The horizontal distance between the base of the ladder and the bottom of the wall, when a 26-foot ladder makes a 78° angle with the ground, is approximately 5.41 feet after using the cosine function and rounding to the nearest hundredth.
Step-by-step explanation:
To determine the horizontal distance between the base of the ladder and the bottom of the wall, we can use trigonometric functions. Specifically, we can use the cosine of the angle that the ladder makes with the ground. The function that correlates the angle, the length of the hypotenuse (ladder), and the adjacent side (horizontal distance) is cosine. In this scenario, we have a 26-foot ladder making a 78° angle with the ground. We can denote the horizontal distance as x.
The cosine of the angle is equal to the adjacent side (x) divided by the hypotenuse (26 feet), so:
cos(78°) = x / 26 ft
Thus,
x = 26 ft * cos(78°)
We'll use a calculator to find the cosine of 78 degrees and multiply it by 26 to find the horizontal distance. Make sure the calculator is set to degrees mode before calculation.
x ≈ 26 ft * 0.2079 (cosine of 78° ≈ 0.2079)
x ≈ 5.4054 feet
After rounding to the nearest hundredth, the horizontal distance between the base of the ladder and the bottom of the wall is approximately 5.41 feet.