Question

how many square feet of outdoor carpet will we need for this hole? 6 ft, 2 ft, 5 ft, 2 ft, 3 ft, 3 ft, 12 ft (description of the image with these measurements)

Explanation:

Step1: Calculate area of large rectangle

The total length is 12 ft and height is 5 ft. Area of rectangle is $length \times height$. So, $12 \times 5 = 60$ square feet.

Step2: Subtract area of the rectangular cut - out

The cut - out has length 2 ft and height 3 ft. Area of cut - out is $2\times3 = 6$ square feet.

Step3: Calculate area of the triangular part (if any) or adjust for the slanted side

Wait, actually, the figure can be considered as a combination. Alternatively, let's re - examine. The left part: from 0 to 6 ft (length 6 ft), height 5 ft, but with a cut - out of 2ft (length) and 3ft (height). Then the right part: from 6 ft to 12 ft (length 6 ft), but the top has a slant. Wait, maybe a better approach:

Total area without any cuts: $12\times5 = 60$.

Subtract the area of the rectangular notch (2ft x 3ft) and also adjust for the triangular part? Wait, no. Wait, the right - hand side: the horizontal length from 6 ft to 12 ft is 6 ft, but the top has a slant. Wait, the horizontal segment on the top right is 2 ft, so the base of the triangle is $6 - 2=4$ ft? Wait, no, maybe I made a mistake.

Wait, let's split the figure into parts:

1. Left rectangle: length 6 ft, height 5 ft: area $6\times5 = 30$. But there is a cut - out of 2ft (length) and 3ft (height): area of cut - out is $2\times3 = 6$. So left part area: $30 - 6=24$.

2. Right part: Let's see, the total length from 6 ft to 12 ft is 6 ft. The height is 5 ft. But the top has a slant. The horizontal length of the top right is 2 ft, so the remaining horizontal length for the triangle is $6 - 2 = 4$ ft? Wait, no. Wait, the vertical height of the slant: the height from the bottom to the top of the slant is 5 ft, and the height of the vertical part on the right is 5 ft. Wait, maybe the right - hand part is a trapezoid. The trapezoid has bases: the bottom base is 6 ft (from x = 6 to x = 12, length 6 ft), the top base is 2 ft, and the height (vertical) is 5 ft? No, trapezoid area is $\frac{(a + b)}{2}\times h$, where a and b are the two parallel sides. Wait, the two parallel sides are the bottom (length 6 ft) and the top (length 2 ft), and the distance between them (height) is 5 ft? No, that's not correct. Wait, maybe the right - hand part is a rectangle with length 6 ft and height 5 ft, minus a triangle. The triangle has base $6 - 2=4$ ft and height 3 ft? Wait, no, the vertical height of the slant: the vertical segment on the right is 5 ft, and the horizontal segment on the top is 2 ft, so the triangle has base $6 - 2 = 4$ ft and height 3 ft? Wait, I think I messed up.

Alternative approach:

Total area of the figure = Area of the big rectangle (12x5) - area of the rectangular cut - out (2x3) - area of the triangular cut - out (if any). Wait, the big rectangle is 12ft (length) x 5ft (height) = 60.

The rectangular cut - out: 2ft (length) x 3ft (height) = 6.

The triangular cut - out: the base of the triangle is $12 - 6 - 2=4$ ft? Wait, no. Wait, the horizontal length from 6 ft to 12 ft is 6 ft. The top horizontal line on the right is 2 ft, so the horizontal difference is $6 - 2 = 4$ ft. The vertical height of the triangle: since the height of the big rectangle is 5 ft, and the vertical segment on the right is 5 ft, but the slant is from the end of the 6 ft mark (x = 6) up to the 2 ft mark on the top (x = 12 - 2=10? Wait, no, the top right has a 2 ft horizontal segment. So from x = 10 to x = 12 (2 ft) is horizontal, and from x = 6 to x = 10 is slanted. The vertical height from the bottom to the top of the slant: the height of the big rectangle is 5 ft, and the height of the vertical part on the left (up to the cut - out) is 5 ft. Wait, maybe the triangle has base $6 - 2 = 4$ ft (horizontal) and height 3 ft (vertical)? Wait, the vertical length of the slant - related triangle: the height of the cut - out on the left is 3 ft, maybe the triangle has height 3 ft? No, this is getting confusing.

Wait, let's use the first method:

Total area = Area of large rectangle (12*5) - area of the small rectangle (2*3) - area of the triangle.

Wait, the triangle: the base is (12 - 6 - 2)=4 ft, and the height is 3 ft? Wait, no, the height of the triangle should be equal to the height of the cut - out? Wait, the height of the big rectangle is 5 ft, and the height of the vertical segment on the left (where the cut - out is) is 3 ft, so the difference in height is 5 - 3 = 2 ft? No, I think I made a mistake.

Wait, let's start over.

The figure can be divided into three parts:

1. Left rectangle: from x = 0 to x = 6, y = 0 to y = 5. But with a cut - out from x = a to x = a + 2, y = 0 to y = 3 (where a is some value, but actually, the cut - out is in the middle of the left 6 ft length). So area of left part: (6*5)-(2*3)=30 - 6 = 24.

2. Middle rectangle: Wait, no. The right - hand part: from x = 6 to x = 12, y = 0 to y = 5. But the top has a slant. The horizontal length from x = 6 to x = 12 is 6 ft. The top horizontal segment is 2 ft (from x = 10 to x = 12), so the base of the triangle is 6 - 2 = 4 ft, and the height of the triangle is 5 - 3 = 2 ft? No, the height of the triangle should be equal to the vertical difference? Wait, no, the vertical height of the triangle is 3 ft? Wait, the vertical length of the left cut - out is 3 ft, maybe the triangle has height 3 ft.

Wait, the area of the right - hand part: area of rectangle (6*5) minus area of triangle ( (4*3)/2 ). Wait, 6*5 = 30, triangle area is (4*3)/2=6. So area of right - hand part: 30 - 6 = 24.

Wait, no, that can't be. Wait, 6*5 is 30, minus triangle area (4*3)/2 = 6, so 30 - 6 = 24. Then total area: 24 (left) + 24 (right)=48? But that doesn't seem right.

Wait, another approach:

Total length = 12 ft, total height = 5 ft.

Area of the figure = (Area of rectangle with length 12 and height 5) - (Area of the rectangular notch (2x3)) - (Area of the triangular notch).

The triangular notch: the base is (12 - 6 - 2)=4 ft (since from 6 ft to 12 ft is 6 ft, and the top has 2 ft horizontal, so the base of the triangle is 6 - 2 = 4 ft), and the height is 3 ft (same as the height of the rectangular notch). So area of triangle is (4*3)/2 = 6.

So total area = (12*5)-(2*3)-(4*3)/2=60 - 6 - 6 = 48.

Wait, let's check:

12*5 = 60 (total area of the big rectangle).

Subtract the area of the small rectangle (2x3 = 6) and the area of the triangle ( (4x3)/2=6 ). So 60 - 6 - 6 = 48.

Yes, that makes sense.

Step1: Calculate area of big rectangle

Area of big rectangle: $12\times5 = 60$ square feet.

Step2: Calculate area of rectangular cut - out

Area of rectangular cut - out: $2\times3=6$ square feet.

Step3: Calculate area of triangular cut - out

The base of the triangle: $12 - 6 - 2 = 4$ feet (length from x = 6 to x = 10, since the top right has 2 ft from x = 10 to x = 12). The height of the triangle is 3 feet (same as the height of the rectangular cut - out). Area of triangle: $\frac{1}{2}\times4\times3 = 6$ square feet.

Step4: Subtract both areas from the big rectangle area

Total area of carpet needed: $60-6 - 6=48$ square feet.

Answer:

48