To ensure safety, the recommended angle that a ladder leaning against a building makes with the ground is 75 degrees. Will a 12 foot ladder reach a window 10 feet above the ground if it leans against the building with this angle? Use trigonometry to justify your answer.
the height the ladder will reach is
12sin75°
To determine if a 12-foot ladder leaning against a building at a 75-degree angle will reach a window 10 feet above the ground, we can use trigonometry to solve for the distance between the base of the ladder and the building.
Let's represent the ladder as the hypotenuse (H), the distance between the base of the ladder and the building as the adjacent side (A), and the height of the window as the opposite side (O) of a right triangle.
We can use the trigonometric ratio of tangent to solve for A (adjacent side) since we know the angle and the opposite side. The formula for tangent is:
tan(angle) = opposite/adjacent
tan(75 degrees) = O/A
Now, we can rearrange the formula to solve for A:
A = O/tan(75 degrees)
Substituting the known values:
A = 10 feet / tan(75 degrees)
To calculate this with a calculator, make sure your calculator is set to degrees mode, then find the tangent of 75 degrees. After that, divide 10 feet by the tangent value to get the value of A.
After calculating, we find that A ≈ 3.8589 feet.
Now, let's determine if the ladder is long enough. We need to measure the total distance from the base of the ladder to the window, which is the sum of A and the ladder's length.
Total distance = A + ladder length
Total distance = 3.8589 feet + 12 feet
Total distance ≈ 15.8589 feet
Since the total distance is approximately 15.8589 feet, the 12-foot ladder will not be long enough to reach the window that is 10 feet above the ground when leaning at a 75-degree angle.