Answer:
Therefore, the tent company needs about 10752 square inches of fabric to make the tent.
Explanation:
To find the surface area of the tent, we need to find the area of each face and add them together. Looking at the net provided, we see that the tent has 5 faces: two congruent triangles (ABD and CDE), two rectangles (ABCF and DEGH), and a parallelogram (BCDE).
Let's start with the triangles. To find the area of triangle ABD, we can use the formula for the area of a triangle:
Area of ABD = (1/2) * base * height
The base is the side AB, which has a length of c = 42 inches, and the height is a, which has length of 67 inches. So we have:
Area of ABD = (1/2) * 42 * 67 = 1407 square inches
Since triangle ABD is congruent to triangle CDE, they have the same area. Therefore, the total area contributed by the two triangles is:
2 * Area of ABD = 2 * 1407 = 2814 square inches
Next, let's find the area of the two rectangles. The rectangle ABCF has length b = 68 inches and height a = 67 inches, so its area is:
Area of ABCF = b * a = 68 * 67 = 4556 square inches
The rectangle DEGH has length d = 64 inches and height a = 67 inches, so its area is:
Area of DEGH = d * a = 64 * 67 = 4288 square inches
Finally, let's find the area of the parallelogram BCDE. To do this, we need to find the height of the parallelogram, which is the perpendicular distance between the base BC and the line DE. We can use the Pythagorean theorem to find this height:
h = sqrt(d^2 - b^2) = sqrt(64^2 - 68^2) ≈ 38.48 inches
Therefore, the area of the parallelogram is:
Area of BCDE = base * height = c * h = 42 * 38.48 ≈ 1613.76 square inches
Adding up the areas of all 5 faces, we get the total surface area of the tent:
Total surface area = 2 * Area of ABD + Area of ABCF + Area of DEGH + Area of BCDE
= 2 * 1407 + 4556 + 4288 + 1613.76 ≈ 10752 square inches
Therefore, the tent company needs about 10752 square inches of fabric to make the tent.