Question

1. soo-jin is installing carpet in a den. using the floorplan below, calculate the area of carpet soo-jin will need to buy.

Explanation:

Step1: Analyze the shape

The floor plan can be considered as a rectangle with a triangle or a pentagon. Let's split it into a rectangle and a triangle. The rectangle has length \( l = 8 \, \text{m} \) and we need to find the width. Wait, actually, looking at the equal marks, the top and bottom sides (excluding the triangle part) are equal, and the left side is 8m. The angle is \( 131.6^\circ \), so the supplementary angle (for the triangle) is \( 180 - 131.6 = 48.4^\circ \)? Wait, maybe better to consider the shape as a rectangle plus a triangle. Wait, the side with 3m: let's assume the rectangle has length \( x \) and width 8m, and the triangle has base 3m and height related to the angle. Wait, maybe another approach: the shape is a pentagon, but we can also think of it as a rectangle with length \( 8 + 3 \cos(180 - 131.6) \)? Wait, no, let's use the fact that the angle is \( 131.6^\circ \), so the horizontal component of the 3m side: \( 3 \cos(180 - 131.6) = 3 \cos(48.4^\circ) \approx 3 \times 0.665 = 1.995 \approx 2 \, \text{m} \)? Wait, no, maybe the length of the rectangle is 8m, and the other part: wait, the left side is 8m, the right side has a 3m segment with angle 131.6 degrees. Let's split the shape into a rectangle and a triangle. The rectangle has length \( L \) and width 8m, and the triangle has two sides: one is 3m, and the angle between the 3m side and the vertical? Wait, maybe the correct way is to extend the sides to form a rectangle and a triangle. Wait, the angle is \( 131.6^\circ \), so the angle between the 3m side and the horizontal (if we consider the bottom side) is \( 180 - 131.6 = 48.4^\circ \). Then the height of the triangle (vertical) is \( 3 \sin(48.4^\circ) \approx 3 \times 0.747 = 2.241 \, \text{m} \), and the base (horizontal) is \( 3 \cos(48.4^\circ) \approx 2 \, \text{m} \). Wait, no, maybe the length of the rectangle is 8m, and the total length of the bottom side is \( 8 + 3 \cos(180 - 131.6) \)? Wait, I think I made a mistake. Let's look at the equal marks: the top, bottom (excluding the triangle), and left side have equal marks, so the top and bottom (excluding the triangle) are equal, and the left side is 8m. So the rectangle part has length \( x \) and width 8m, and the triangle part has a base of 3m and the angle at the bottom is \( 131.6^\circ \), so the height of the triangle (vertical) is \( 3 \sin(180 - 131.6) = 3 \sin(48.4^\circ) \approx 2.24 \, \text{m} \), and the horizontal component is \( 3 \cos(48.4^\circ) \approx 2 \, \text{m} \). Wait, maybe the correct approach is to calculate the area as the area of the rectangle plus the area of the triangle. Wait, no, actually, the shape is a rectangle with a triangle attached? Wait, no, looking at the diagram, it's a pentagon with two equal sides (top and bottom, left side 8m, right side has a 3m segment with angle 131.6 degrees). Let's use the formula for the area of a polygon. Alternatively, split the shape into a rectangle and a triangle. The rectangle has length \( 8 \, \text{m} \) and width \( 8 \, \text{m} \)? No, the left side is 8m. Wait, maybe the length of the rectangle is \( 8 \, \text{m} \) and the width is \( 8 \, \text{m} \), and the triangle has a base of \( 3 \, \text{m} \) and height calculated from the angle. Wait, the angle is \( 131.6^\circ \), so the angle between the 3m side and the vertical is \( 131.6 - 90 = 41.6^\circ \)? No, that's not right. Wait, let's use trigonometry. The angle given is \( 131.6^\circ \), so the adjacent side (horizontal) to this angle is \( 3 \cos(131.6^\circ) \), but since cosine is negative in the second quadrant, \( \cos(131.6^\circ) = -\cos(48.4^\circ) \approx -0.665 \), so the horizontal component is \( -3 \times 0.665 \approx -2 \, \text{m} \), meaning it's 2m to the left. The vertical component is \( 3 \sin(131.6^\circ) = 3 \sin(48.4^\circ) \approx 2.24 \, \text{m} \). Wait, maybe the shape is a rectangle with length \( 8 + 2 = 10 \, \text{m} \) and width \( 8 \, \text{m} \), minus a triangle? No, this is getting confusing. Wait, another way: the area can be calculated as the area of a rectangle plus the area of a triangle. Let's assume the rectangle has length \( 8 \, \text{m} \) and width \( 8 \, \text{m} \), and the triangle has a base of \( 3 \, \text{m} \) and height \( 8 \, \text{m} \)? No, that doesn't make sense. Wait, looking at the equal marks, the top, bottom, and left side are equal, so the top and bottom are 8m? No, the left side is 8m. Wait, maybe the length of the rectangle is \( 8 \, \text{m} \) and the width is \( 8 \, \text{m} \), and the triangle has a base of \( 3 \, \text{m} \) and the angle is \( 131.6^\circ \), so the area of the triangle is \( \frac{1}{2} \times 3 \times 8 \times \sin(131.6^\circ) \)? Wait, no, the angle between the two sides of the triangle: if one side is 3m and the other is 8m, then the area is \( \frac{1}{2}ab \sin\theta \). Wait, maybe the shape is a rectangle with length 8m and width 8m, plus a triangle with sides 3m and 8m and angle 131.6 degrees. Let's calculate that. The area of the rectangle is \( 8 \times 8 = 64 \, \text{m}^2 \). The area of the triangle is \( \frac{1}{2} \times 3 \times 8 \times \sin(131.6^\circ) \). \( \sin(131.6^\circ) = \sin(180 - 48.4) = \sin(48.4^\circ) \approx 0.747 \). So the area of the triangle is \( \frac{1}{2} \times 3 \times 8 \times 0.747 \approx 12 \times 0.747 \approx 8.964 \, \text{m}^2 \). Then total area is \( 64 + 8.964 \approx 72.964 \, \text{m}^2 \). Wait, but that doesn't seem right. Wait, maybe the length of the rectangle is not 8m. Wait, the left side is 8m, and the top and bottom sides (excluding the triangle) are equal, so let's say the length of the rectangle is \( x \), and the triangle has a base of 3m, and the angle is 131.6 degrees. Then the total area is \( x \times 8 + \frac{1}{2} \times 3 \times 8 \times \sin(131.6^\circ) \). But we need to find \( x \). Wait, maybe the top and bottom sides (excluding the triangle) are equal to the left side? No, the left side is 8m, and the top and bottom have equal marks, so they are equal. Wait, maybe the length of the rectangle is 8m, and the triangle is attached to the right side. Wait, the angle is 131.6 degrees, so the horizontal component of the 3m side is \( 3 \cos(131.6^\circ) \approx -2 \, \text{m} \) (meaning 2m to the left), so the total length of the bottom side is \( 8 - 2 = 6 \, \text{m} \)? No, this is too confusing. Wait, let's use the formula for the area of a pentagon, but maybe it's a rectangle with a triangle. Wait, another approach: the shape can be divided into a rectangle and a triangle. The rectangle has length \( 8 \, \text{m} \) and width \( 8 \, \text{m} \), and the triangle has a base of \( 3 \, \text{m} \) and height \( 8 \, \text{m} \), but the angle is 131.6 degrees. Wait, no, the angle between the 3m side and the 8m side is 131.6 degrees. So the area of the triangle is \( \frac{1}{2} \times 3 \times 8 \times \sin(131.6^\circ) \approx \frac{1}{2} \times 24 \times 0.747 \approx 8.96 \, \text{m}^2 \). The area of the rectangle is \( 8 \times 8 = 64 \, \text{m}^2 \). So total area is \( 64 + 8.96 = 72.96 \, \text{m}^2 \), approximately 73 \( \text{m}^2 \). Wait, but maybe the length of the rectangle is not 8m. Wait, the left side is 8m, and the top and bottom sides (excluding the triangle) are equal, so let's say the length of the rectangle is \( 8 \, \text{m} \), and the triangle is attached to the right end. So the total area is the area of the rectangle plus the area of the triangle. Alternatively, maybe the shape is a square with a triangle, but the square has side 8m, and the triangle has base 3m and height 8m with angle 131.6 degrees. Wait, I think I made a mistake. Let's check the angle: \( 131.6^\circ \), so \( \sin(131.6^\circ) = \sin(48.4^\circ) \approx 0.747 \), \( \cos(131.6^\circ) = -\cos(48.4^\circ) \approx -0.665 \). So the horizontal component of the 3m side is \( 3 \times (-0.665) \approx -2 \, \text{m} \) (so 2m to the left), and the vertical component is \( 3 \times 0.747 \approx 2.24 \, \text{m} \). So the total length of the bottom side is \( 8 - 2 = 6 \, \text{m} \)? No, that can't be. Wait, maybe the correct way is to consider the shape as a rectangle with length \( 8 + 3 \cos(180 - 131.6) \)? Wait, \( 180 - 131.6 = 48.4 \), so \( \cos(48.4) \approx 0.665 \), so \( 3 \times 0.665 \approx 2 \), so the length of the rectangle is \( 8 + 2 = 10 \, \text{m} \), and the width is 8m, and then subtract the area of the triangle? No, this is too confusing. Wait, maybe the answer is 72 \( \text{m}^2 \) or 73 \( \text{m}^2 \). Wait, let's try again. The shape is a pentagon, which can be divided into a rectangle and a triangle. The rectangle has length \( 8 \, \text{m} \) and width \( 8 \, \text{m} \), area \( 64 \, \text{m}^2 \). The triangle has two sides: 3m and 8m, and the included angle is \( 131.6^\circ \). The area of a triangle with two sides \( a \) and \( b \) and included angle \( \theta \) is \( \frac{1}{2}ab \sin\theta \). So \( \frac{1}{2} \times 3 \times 8 \times \sin(131.6^\circ) \approx \frac{1}{2} \times 24 \times 0.747 \approx 8.96 \, \text{m}^2 \). So total area is \( 64 + 8.96 = 72.96 \approx 73 \, \text{m}^2 \). Alternatively, maybe the length of the rectangle is 8m, and the width is 8m, and the triangle is attached to the right side, so the total area is \( 8 \times 8 + \frac{1}{2} \times 3 \times 8 \times \sin(131.6^\circ) \approx 64 + 9 = 73 \, \text{m}^2 \).

Step2: Calculate the area

Wait, maybe I was wrong. Let's look at the diagram again. The left side is 8m, the top and bottom sides (excluding the triangle) are equal, and the right side has a 3m segment with angle 131.6 degrees. So the shape is a rectangle with length \( L \) and width 8m, and a triangle with base 3m and height \( h \). The angle between the 3m side and the vertical is \( 131.6 - 90 = 41.6^\circ \)? No, that's not. Wait, the angle is between the 3m side and the horizontal (bottom side), so it's 131.6 degrees, so the height (vertical) of the triangle is \( 3 \sin(131.6^\circ) \approx 2.24 \, \text{m} \), and the base (horizontal) is \( 3 \cos(131.6^\circ) \approx -2 \, \text{m} \) (so 2m to the left). So the length of the rectangle is \( 8 - 2 = 6 \, \text{m} \)? No, that would make the rectangle area \( 6 \times 8 = 48 \, \text{m}^2 \), and the triangle area \( \frac{1}{2} \times 3 \times 8 \times \sin(131.6^\circ) \approx 9 \, \text{m}^2 \), total \( 57 \, \text{m}^2 \), which is not right. Wait, I think the correct approach is to realize that the shape is a rectangle with length 8m and width 8m, plus a triangle with base 3m and height 8m, but the angle is 131.6 degrees. Wait, no, the angle is between the 3m side and the 8m side, so the area of the triangle is \( \frac{1}{2} \times 3 \times 8 \times \sin(131.6^\circ) \approx 9 \, \text{m}^2 \), and the rectangle is \( 8 \times 8 = 64 \, \text{m}^2 \), so total \( 73 \, \text{m}^2 \). I think that's the answer.

Answer:

\boxed{73} (approximate, depending on the exact calculation)