Use the image to answer the question.
An illustration shows a wing-like structure with two-part wings on either side, and a magnified, labeled figure of a two-part wing. The magnified two-part wing shows two similar pentagons, stacked in opposite directions upon each other, sharing a common base. The description of the top pentagon is as follows: The base is 1.4 meters, the bottom left and right sides are smaller, and the top left and right sides are longer. A dashed line joining the top left and right vertices is labeled as 2.6 meters. The triangular region above this dashed line is shaded. The perpendicular distance from the top vertex to the dashed line is 0.3 meters, and from the dashed line to the base is 0.4 meters. The base is extended out to the right to the length of the top right vertex using a dashed line, labeled as 0.6 meters. The right vertices on the top and bottom pentagons are joined with a solid line labeled as height 0.8 meters. This line forms a triangle with the right short sides of the pentagons. This triangle is shaded. Adjacent to magnified wing, a winged figure is shown with a long central oval shape with two-part wings extending left and right from two small triangles above the long oval. A small circle sits on top of the figure.
You are making a dragonfly kite. You want to cover all parts of all four wings, including the triangles in the center, with shiny nylon. The dimensions of one pair of wings is shown. The kite will need two sets of wings. Decompose the figure on the left into simple shapes. How many square meters of nylon will you need to cover both pairs of wings?
(1 point)
Responses
5.16 square meters
5.16 square meters
10.48 square meters
10.48 square meters
5.24 square meters
5.24 square meters
9.52 square meters
9.52 square meters
The magnified wing can be decomposed into a rectangle with base 1.4 meters and height 0.3 meters, and a trapezoid with bases of 0.4 meters and 2.6 meters and height 0.8 meters. The area of the rectangle is 0.42 square meters and the area of the trapezoid is ((0.4 + 2.6) / 2) * 0.8 = 1.6 square meters. So, the total area of one set of wings is 2 * (2 * 0.42 + 1.6) = 5.16 square meters. The kite will need two sets, so the total area needed is 2 * 5.16 = 10.32 square meters.
The closest answer is 10.48 square meters, so the correct response is:
10.48 square meters.
To decompose the figure into simple shapes, we can break down the illustration as follows:
1. The top pentagon:
- It consists of a trapezoid (the shaded triangular region above the dashed line) and a rectangle (the remaining bottom part).
- The trapezoid has a base of 1.4 meters, a height of 0.4 meters, and a perpendicular distance of 0.3 meters.
- The rectangle has a length of 1.4 meters and a width of 0.8 meters.
2. The bottom pentagon:
- It consists of a rectangle (the shaded triangular region formed by the bottom part) and a larger rectangle (the remaining top part).
- The smaller rectangle has a length of 1.4 meters and a width of 0.4 meters.
- The larger rectangle has a length of 1.4 meters and a width of 0.8 meters.
To find the total area of nylon needed to cover both pairs of wings, we add up the areas of the individual shapes:
Area of trapezoid = (base1 + base2) * height / 2 = (1.4 + 0.6) * 0.4 / 2 = 0.8 square meters
Area of rectangle (top of top pentagon) = length * width = 1.4 * 0.8 = 1.12 square meters
Area of rectangle (bottom of top pentagon) = length * width = 1.4 * 0.4 = 0.56 square meters
Area of rectangle (shaded triangular region formed by the bottom part) = length * width = 1.4 * 0.4 = 0.56 square meters
Area of larger rectangle (remaining top part of bottom pentagon) = length * width = 1.4 * 0.8 = 1.12 square meters
Area of smaller rectangle (shaded triangular region formed by the bottom part) = length * width = 1.4 * 0.4 = 0.56 square meters
Total area = Area of trapezoid + Area of top rectangle of top pentagon + Area of bottom rectangle of top pentagon + Area of shaded triangular region (bottom part) + Area of larger rectangle (bottom part of bottom pentagon) + Area of smaller rectangle (bottom part of bottom pentagon)
= 0.8 + 1.12 + 0.56 + 0.56 + 1.12 + 0.56
= 5.12 square meters
Therefore, you will need 5.12 square meters of nylon to cover both pairs of wings. However, none of the given answer choices match this result.
To decompose the figure on the left into simple shapes, we can break it down into rectangles and triangles.
First, let's calculate the area of the rectangular part of each wing. The rectangular part has a base of 1.4 meters and a height of 0.8 meters. So the area of one rectangular part is:
Area of rectangular part = base * height = 1.4 * 0.8 = 1.12 square meters
For each pair of wings, there are two rectangular parts. So the total area of the rectangular parts for both pairs of wings is:
Total area of rectangular parts = 2 * 1.12 = 2.24 square meters
Next, let's calculate the area of each triangular part of each wing. The triangular part has a base of 0.6 meters and a height of 0.4 meters. So the area of one triangular part is:
Area of triangular part = 0.5 * base * height = 0.5 * 0.6 * 0.4 = 0.12 square meters
For each pair of wings, there are two triangular parts. So the total area of the triangular parts for both pairs of wings is:
Total area of triangular parts = 2 * 0.12 = 0.24 square meters
Now, let's calculate the area of the shaded region above the dashed line for one wing. It is formed by subtracting the area of the triangle (0.12 square meters) from the area of the rectangle (1.12 square meters):
Area of shaded region = Area of rectangular part - Area of triangular part = 1.12 - 0.12 = 1.0 square meters
For each pair of wings, there are two shaded regions. So the total area of the shaded regions for both pairs of wings is:
Total area of shaded regions = 2 * 1.0 = 2.0 square meters
Finally, the total area of nylon needed to cover both pairs of wings is the sum of the areas of the rectangular parts, triangular parts, and shaded regions:
Total area of nylon needed = Total area of rectangular parts + Total area of triangular parts + Total area of shaded regions
= 2.24 + 0.24 + 2.0
= 4.48 square meters
Therefore, you will need 4.48 square meters of nylon to cover both pairs of wings.
Since none of the provided answer choices match, it seems there might be an error in the calculations or the information provided. I would recommend double-checking the dimensions and recalculating the areas to get the correct answer.