Question

5 feet 5 feet 5 feet 5 feet house 10 feet 15 feet porch 30 feet

Explanation:

Step1: Find the length of the top base of the house

The bottom base of the house (the horizontal side) is 30 feet. The two sides on the top (each 5 feet) are cut off, so the top base length is \( 30 - 5 - 5 = 20 \) feet.

Step2: Calculate the area of the house (a trapezoid)

The formula for the area of a trapezoid is \( A=\frac{(b_1 + b_2)h}{2} \), where \( b_1 = 30 \) feet, \( b_2 = 20 \) feet, and \( h = 10 \) feet.
\[ A_{house}=\frac{(30 + 20)\times10}{2}=\frac{50\times10}{2}=250 \] square feet.

Step3: Calculate the area of the porch (a rectangle minus a trapezoid? Wait, no. The porch is a rectangle with length 30 feet and height 15 feet, minus the area of the trapezoid - shaped indentation. Wait, actually, the indentation on the porch is a trapezoid. Let's find the area of the porch. The porch's main part is a rectangle of \( 30\times15 \), but we subtract the area of the trapezoid indentation. Wait, no, looking at the diagram, the house is above the porch. Wait, maybe the house is a trapezoid (top part) and the porch is a rectangle with a trapezoid - shaped cut - out? Wait, no, let's re - examine. The total figure: the house has a top part (a trapezoid) with height 10 feet, and the porch is a rectangle (30x15) with a trapezoid - shaped indentation (the same as the top of the house? Wait, no. Wait, the house's bottom base is 30, top base is 20, height 10. The porch: the vertical height is 15, and the horizontal length is 30. But there is a trapezoid - shaped cut - out on the porch (the part under the house's bottom trapezoid). Wait, no, maybe the house is a trapezoid (area calculated above) and the porch is a rectangle (30x15) minus the area of the trapezoid that is the same as the house's top? No, wait, actually, the porch is a rectangle with length 30 and height 15, and the indentation is a trapezoid with bases 30 and 20 (same as the house's bases) and height equal to the height of the indentation? Wait, no, the height of the indentation: the vertical distance from the bottom of the house to the top of the porch's indentation. Wait, maybe I made a mistake. Let's start over.

Wait, maybe the house is a trapezoid (top base 20, bottom base 30, height 10) and the porch is a rectangle (30x15) minus the area of the trapezoid (top base 20, bottom base 30, height equal to the height of the indentation). Wait, the height of the indentation: the vertical length from the bottom of the house (where it meets the porch) to the top of the porch's indentation. Wait, the house's height is 10, the porch's height is 15. Wait, no, the diagram shows the house has a height of 10 feet (vertical) and the porch has a height of 15 feet (vertical). The horizontal length is 30 feet.

Wait, another approach: the total area of the figure (house + porch) can be calculated as the area of the large rectangle (30x(10 + 15)) plus the area of the two triangles on the top of the house. Wait, the top of the house has two right triangles, each with base 5 and height 5.

Step4: Calculate the area of the two triangles on the top of the house

Each triangle has area \( \frac{1}{2}\times5\times5 = 12.5 \) square feet. Two triangles: \( 2\times12.5 = 25 \) square feet.

Step5: Calculate the area of the large rectangle (house + porch without the top triangles)

The large rectangle has length 30 and height \( 10 + 15=25 \) feet. Area \( A_{rectangle}=30\times25 = 750 \) square feet.

Step6: Calculate the total area (house + porch)

Add the area of the two triangles to the area of the rectangle: \( 750+25 = 775 \) square feet. Wait, no, that's not correct. Wait, the house's top part: the two triangles and the middle rectangle. The middle rectangle of the house has length 20 and height 10, and the two triangles have base 5 and height 5 (wait, no, the height of the triangles is 5, and the base is 5? Wait, the vertical height from the top of the house to the horizontal line is 5 feet, and the horizontal length of each triangle is 5 feet. So each triangle is a right triangle with legs 5 and 5. So area of each triangle is \( \frac{1}{2}\times5\times5 = 12.5 \), two triangles: 25. The middle rectangle of the house: length 20, height 10, area \( 20\times10 = 200 \). Then the house's area is \( 200 + 25=225 \)? Wait, no, earlier I calculated the trapezoid area as 250. There is a mistake here. Let's recalculate the trapezoid area. The formula for trapezoid is \( \frac{(a + b)h}{2} \), where \( a = 30 \), \( b = 20 \), \( h = 10 \). \( \frac{(30 + 20)\times10}{2}=250 \). The two triangles: if we consider the top of the house, the total top length (before cutting) would be \( 20+5 + 5=30 \), so the house's top is a trapezoid (30 and 20 as the two bases, height 10) and the two triangles are on the sides? No, maybe the house is a pentagon? Wait, the diagram shows the house has a top with two right - angled sides (each 5 feet horizontal and 5 feet vertical) and a middle horizontal side. So the house can be divided into a rectangle (20x10) and two right triangles (each 5x5). So area of rectangle: \( 20\times10 = 200 \), area of two triangles: \( 2\times\frac{1}{2}\times5\times5 = 25 \), total house area: \( 200 + 25=225 \). Ah, here is the mistake earlier. The height of the triangles is 5, not 10. The vertical height from the top of the house to the horizontal line (the top of the middle rectangle) is 5 feet. So the trapezoid's height is 10 feet? No, the vertical distance between the two parallel sides (the bottom base of the house - 30 feet and the top base - 20 feet) is 10 feet. The two triangles are on the top, with vertical height 5 feet and horizontal base 5 feet. So the house is composed of a rectangle (20x10) and two right triangles (each with base 5 and height 5). Wait, no, the vertical height of the triangles is 5, and the horizontal length of the triangles is 5. So the total height of the house (from the bottom of the house to the top of the triangles) is \( 10+5 = 15 \)? No, the diagram shows the house has a vertical height of 10 feet (the distance from the porch to the top of the house's main part) and the triangles are on top of that? I think I misinterpreted the diagram. Let's start over with the correct division:

1. **House Area**:
- The house is a trapezoid with bases \( b_1 = 30 \) ft and \( b_2=30 - 5 - 5 = 20 \) ft, and height \( h = 10 \) ft.
- Area of trapezoid: \( A_{house}=\frac{(30 + 20)\times10}{2}=250 \) square feet.
2. **Porch Area**:
- The porch is a rectangle with length \( l = 30 \) ft and height \( h = 15 \) ft, minus the area of the trapezoid - shaped indentation (which is the same as the area of the house's trapezoid? No, wait, the indentation on the porch is a trapezoid with bases \( b_1 = 30 \) ft and \( b_2 = 20 \) ft, and height equal to the height of the indentation. Wait, no, the porch's indentation is a trapezoid with the same bases as the house's trapezoid and height equal to the height of the indentation. But looking at the diagram, the vertical distance from the bottom of the house to the top of the porch's indentation is the same as the house's height? No, this is getting too complicated. Let's use the correct method:

The figure can be divided into:

- A rectangle for the porch: \( 30\times15 = 450 \) square feet.
- A trapezoid for the house: with bases 30 and \( 30 - 5 - 5=20 \), height 10. Area \( \frac{(30 + 20)\times10}{2}=250 \) square feet.
- Two right triangles on the top of the house: each with legs 5 and 5. Area of each triangle: \( \frac{1}{2}\times5\times5 = 12.5 \), two triangles: \( 2\times12.5 = 25 \) square feet.

Wait, no, the two triangles are part of the house. So total area of the house and porch:

\( A = 450+250 + 25=725 \)? No, that's not right. Wait, the correct way:

The house has a top part which is a trapezoid (bases 30 and 20, height 10) and two triangles (each 5x5). The porch is a rectangle (30x15).

Wait, let's calculate the total area:

- Porch: \( 30\times15=450 \)
- House (trapezoid): \( \frac{(30 + 20)\times10}{2}=250 \)
- Two triangles: \( 2\times\frac{1}{2}\times5\times5 = 25 \)
Total area: \( 450+250 + 25 = 725 \) square feet.

Wait, but maybe the two triangles are not part of the house but are extra? No, the diagram shows the house has a pentagonal shape (the top has two triangles and a rectangle). So the house's area is the area of the rectangle (20x10) plus the area of the two triangles (2x(0.5x5x5)): \( 20\times10+25 = 225 \), and the porch is a rectangle (30x15) minus the area of the trapezoid (which is the same as the house's rectangle? No, this is confusing. Let's use the initial trapezoid for the house:

House area (trapezoid): \( \frac{(30 + 20)\times10}{2}=250 \)

Porch area: The porch is a rectangle (30x15) minus the area of the trapezoid - shaped indentation. The indentation is a trapezoid with bases 30 and 20, and height equal to the height of the indentation. Wait, the height of the indentation is the same as the house's height? No, the house's height is 10, so the indentation's height is 10? No, the porch's height is 15, so the indentation's height is 10? This is incorrect.

Let's start over with the correct diagram interpretation:

- The bottom part (porch) is a rectangle with length 30 ft and height 15 ft. Area: \( 30\times15 = 450 \) sq ft.
- The middle part (house's main body) is a rectangle with length 30 ft and height 10 ft. But there are two triangular cut - outs on the top of this rectangle, each with base 5 ft and height 5 ft. Wait, no, the top of the house has two triangular extensions, not cut - outs. So the house's main body is a rectangle (30x10), and on top of it, there are two right triangles (each with base 5 and height 5). So the house's area is \( 30\times10+2\times\frac{1}{2}\times5\times5=300 + 25 = 325 \) sq ft.
- Then the total area (house + porch) is \( 450+325 = 775 \) sq ft.

Ah, this makes more sense. The house has a rectangular part (30x10) and two triangular extensions (each 5x5). So:

- Area of rectangular part of house: \( 30\times10 = 300 \)
- Area of two triangles: \( 2\times\frac{1}{2}\times5\times5 = 25 \)
- Area of porch: \( 30\times15 = 450 \)
- Total area: \( 300+25 + 450=775 \) square feet.

Answer:

The total area of the house and porch is 775 square feet.