Final answer:
To find the horizontal distance from the base of the skyscraper to the ship, we can use the tangent function with the angle of depression and the height of the observation deck. By rearranging the equation and calculating the tangent of the angle, we can determine the horizontal distance.
Step-by-step explanation:
To find the horizontal distance from the base of the skyscraper to the ship, we can use trigonometry. Let's draw a diagram to represent the situation. The angle of depression is 42 degrees, which means the angle between the observation deck and the horizontal line connecting the base of the skyscraper and the ship. The observation deck is 872 feet high.
We can use the tangent function to find the horizontal distance. The tangent of the angle of depression is equal to the opposite side (the height of the observation deck) divided by the adjacent side (the horizontal distance). Using the formula tan(angle) = opposite/adjacent, we get tan(42) = 872/adjacent.
Now, we can solve for the adjacent side by rearranging the equation. Multiply both sides by the adjacent side and divide by tan(42). The formula becomes adjacent = 872/tan(42). Use a calculator to find the value of tan(42) and then divide 872 by that value to get the horizontal distance from the base of the skyscraper to the ship.