A contractor building eight ocean-front condominiums wants to maximize the view of the ocean for each unit. The side of the building facing the ocean is not built in a straight line parallel to the ocean, but instead is built in a zigzag pattern as shown in the figure. Each condo has a window of length x facing the ocean at an angle of 70° from a line perpendicular to the ocean.
b. Find the length x of each window. Round to the nearest foot. The windows facing the ocean are 8 ft high and x feet wide. By using the zigzag pattern how much more ocean-front viewing area does each window provide than if the windows were parallel to the ocean? Round to the nearest square foot.

Question

29-03-2024
Mathematics

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A contractor building eight ocean-front condominiums wants to maximize the view of the ocean for each unit. The side of the building facing the ocean is not built in a straight line parallel to the ocean, but instead is built in a zigzag pattern as shown in the figure. Each condo has a window of length x facing the ocean at an angle of 70° from a line perpendicular to the ocean.
b. Find the length x of each window. Round to the nearest foot. The windows facing the ocean are 8 ft high and x feet wide. By using the zigzag pattern how much more ocean-front viewing area does each window provide than if the windows were parallel to the ocean? Round to the nearest square foot.

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akar69 akar69 · Answer 1 · 2024-03-29T05:41:00+01:00

To find the length x of each window, we can use trigonometry. The given angle of 70° is with respect to the line perpendicular to the ocean.

1. We can consider the right triangle formed by the window's height (8 ft), the window's width (x ft), and the angle of 70°.

2. We know that the tangent function relates the opposite side to the adjacent side in a right triangle. So, we have:

.

3. Solving for x, we get:

.

Therefore, the length x of each window is approximately 3 feet.

To find how much more ocean-front viewing area each window provides in the zigzag pattern compared to if the windows were parallel to the ocean:

1. The viewing area in the zigzag pattern is in the shape of a trapezoid. The formula for the area of a trapezoid is:

, where a and b are the lengths of the parallel sides and h is the height.

2. In this case, one side of the trapezoid (a) is the width x of the window, the other side (b) can be calculated as

, and the height (h) is 8 ft.

3. Calculate the areas for both scenarios and find the difference to determine how much more ocean-front viewing area each window provides in the zigzag pattern compared to if the windows were parallel to the ocean.

By following these steps, you can accurately calculate the length of each window and determine the additional ocean-front viewing area provided by the zigzag pattern compared to parallel windows.