Question

find the area of the figure. use 3.14 for π. the area of the figure is \boxed{} ft².

Explanation:

Step1: Analyze the figure's components

The figure can be divided into three parts: a rectangle, a right triangle on the left, and a right triangle on the right. The rectangle has dimensions \(4\,\text{ft}\) (width) and \(8\,\text{ft}\) (height). The left triangle has a base of \(3\,\text{ft}\) and height of \(3\,\text{ft}\). The right triangle has a base of \(2\,\text{ft}\) and height of \(3\,\text{ft}\).

Step2: Calculate the area of the rectangle

The formula for the area of a rectangle is \(A = l\times w\), where \(l\) is the length and \(w\) is the width. For the rectangle, \(l = 8\,\text{ft}\) and \(w = 4\,\text{ft}\). So, \(A_{\text{rectangle}} = 8\times4 = 32\,\text{ft}^2\).

Step3: Calculate the area of the left triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times b\times h\), where \(b\) is the base and \(h\) is the height. For the left triangle, \(b = 3\,\text{ft}\) and \(h = 3\,\text{ft}\). So, \(A_{\text{left triangle}}=\frac{1}{2}\times3\times3=\frac{9}{2}= 4.5\,\text{ft}^2\).

Step4: Calculate the area of the right triangle

For the right triangle, \(b = 2\,\text{ft}\) and \(h = 3\,\text{ft}\). Using the triangle area formula, \(A_{\text{right triangle}}=\frac{1}{2}\times2\times3 = 3\,\text{ft}^2\).

Step5: Sum the areas of all components

To find the total area of the figure, we add the areas of the rectangle, left triangle, and right triangle. \(A_{\text{total}}=A_{\text{rectangle}} + A_{\text{left triangle}}+A_{\text{right triangle}}\). Substituting the values, we get \(A_{\text{total}} = 32 + 4.5+ 3=39.5\,\text{ft}^2\)? Wait, no, wait. Wait, maybe I misread the figure. Wait, looking again, maybe the height of the triangles is related to the vertical side. Wait, actually, maybe the figure is a combination where the top part is a trapezoid? Wait, no, the original figure: the rectangle is \(4\) ft wide and \(8\) ft tall. Then above the rectangle, there are two triangles? Wait, no, maybe the left triangle has base \(3\) and height \(3\), right triangle base \(2\) and height \(3\), and the middle part? Wait, no, maybe the total width of the top is \(3 + 4+ 2\)? Wait, no, the figure: let's re - examine. The rectangle is \(4\) ft (width) and \(8\) ft (height). Then above the rectangle, there is a shape that can be split into two triangles. Wait, maybe the left triangle: base \(3\) ft, height \(3\) ft. Right triangle: base \(2\) ft, height \(3\) ft. And the rectangle is \(4\) ft by \(8\) ft. Wait, but also, maybe the height of the triangles is \(3\) ft, and the rectangle is \(4\) ft by \(8\) ft. Wait, but when we calculate, \(8\times4 = 32\), left triangle: \(\frac{1}{2}\times3\times3 = 4.5\), right triangle: \(\frac{1}{2}\times2\times3=3\). Then total area is \(32 + 4.5+3 = 39.5\)? But that seems low. Wait, maybe I made a mistake in the figure's components. Wait, maybe the rectangle is \(4\) ft (width) and \(8 + 3=11\) ft? No, the figure shows the vertical sides: the rectangle has height \(8\) ft, and the triangles have height \(3\) ft above the rectangle? Wait, no, the dashed lines show that the height of the triangles is \(3\) ft, and the rectangle is \(8\) ft tall. Wait, maybe the correct way is: the figure is composed of a rectangle (\(4\times8\)), a left triangle (\(3\times3\) base and height), and a right triangle (\(2\times3\) base and height). Wait, but let's recalculate:

Wait, another approach: the figure can be seen as a rectangle with length \(8\) and width \(4\), plus a left triangle with base \(3\) and height \(3\), plus a right triangle with base \(2\) and height \(3\). So:

\(A_{\text{rectangle}}=4\times8 = 32\)

\(A_{\text{left triangle}}=\frac{1}{2}\times3\times3 = 4.5\)

\(A_{\text{right triangle}}=\frac{1}{2}\times2\times3 = 3\)

Total area \(=32 + 4.5+3=39.5\)? But that seems incorrect. Wait, maybe the height of the triangles is the same as the width of the rectangle? No, the dashed lines show the height of the triangles is \(3\) ft. Wait, maybe I misread the base of the left triangle. Wait, the left triangle's base is \(3\) ft, height \(3\) ft. Right triangle: base \(2\) ft, height \(3\) ft. Rectangle: \(4\) ft (width) and \(8\) ft (height). Wait, but maybe the figure is a trapezoid on top of a rectangle? Wait, the top part: the two triangles and the rectangle's top? No, the rectangle is \(4\) ft wide, so the top of the rectangle is \(4\) ft. Then the left triangle is attached to the left side of the rectangle's top, and the right triangle to the right side. So the total width of the figure at the top is \(3 + 4+ 2=9\) ft? Wait, no, the left triangle has a base of \(3\) ft (horizontal), right triangle base \(2\) ft (horizontal), and the rectangle is \(4\) ft (horizontal). So the total horizontal length at the top is \(3 + 4+ 2 = 9\) ft, and the height of the top part (the triangles) is \(3\) ft, and the rectangle is \(8\) ft tall. Wait, no, maybe the figure is composed of a rectangle (\(4\times8\)) and a trapezoid on top. The trapezoid has bases \(4\) ft and \(3 + 4+ 2=9\) ft? No, that doesn't make sense. Wait, let's use the initial approach. Wait, maybe I miscalculated. Wait, the left triangle: base \(3\) ft, height \(3\) ft. Area: \(\frac{1}{2}\times3\times3 = 4.5\). Right triangle: base \(2\) ft, height \(3\) ft. Area: \(\frac{1}{2}\times2\times3 = 3\). Rectangle: \(4\times8 = 32\). Then total area: \(32+4.5 + 3=39.5\)? But that seems wrong. Wait, maybe the height of the triangles is \(3\) ft, and the rectangle is \(4\) ft by \(8\) ft, but the triangles are attached to the top of the rectangle, so the total height of the figure is \(8 + 3=11\) ft? No, the figure shows the vertical sides: the rectangle has height \(8\) ft, and the triangles have height \(3\) ft (dashed lines are vertical). Wait, I think I made a mistake in the components. Let's look again. The figure: there is a rectangle with width \(4\) ft and height \(8\) ft. Then, above the rectangle, on the left, there is a right triangle with base \(3\) ft (horizontal) and height \(3\) ft (vertical). On the right, a right triangle with base \(2\) ft (horizontal) and height \(3\) ft (vertical). So the total area is the area of the rectangle plus the area of the two triangles. So:

\(A_{\text{rectangle}}=4\times8 = 32\)

\(A_{\text{left triangle}}=\frac{1}{2}\times3\times3 = 4.5\)

\(A_{\text{right triangle}}=\frac{1}{2}\times2\times3 = 3\)

Total area \(=32 + 4.5+3=39.5\)? Wait, but that seems too small. Wait, maybe the height of the triangles is \(3\) ft, but the base of the left triangle is \(3\) ft (vertical)? No, the dashed lines are horizontal? Wait, the right angle symbols: the left triangle has a right angle, so the base is horizontal (3 ft) and height is vertical (3 ft). The right triangle also has a right angle, base horizontal (2 ft) and height vertical (3 ft). The rectangle has width 4 ft (horizontal) and height 8 ft (vertical). So the total area should be \(32+4.5 + 3=39.5\) square feet? Wait, no, maybe I misread the dimensions. Wait, the left triangle: base 3 ft, height 3 ft. Right triangle: base 2 ft, height 3 ft. Rectangle: 4 ft (width) and 8 ft (height). Wait, but maybe the figure is a combination where the two triangles and the rectangle form a larger shape. Wait, another way: the figure can be considered as a trapezoid with bases \(4\) ft and \(3 + 4+ 2 = 9\) ft, and height \(3\) ft, plus a rectangle with dimensions \(4\) ft and \(8\) ft. The area of a trapezoid is \(\frac{(a + b)}{2}\times h\), where \(a\) and \(b\) are the bases, and \(h\) is the height. So for the trapezoid, \(a = 4\), \(b = 9\), \(h = 3\). Area of trapezoid: \(\frac{(4 + 9)}{2}\times3=\frac{13}{2}\times3 = 19.5\,\text{ft}^2\). Area of rectangle: \(4\times8 = 32\,\text{ft}^2\). Total area: \(19.5+32 = 51.5\,\text{ft}^2\). Ah! Here's the mistake. I split the top part into two triangles, but it's actually a trapezoid. The top base of the trapezoid is \(3 + 4+ 2=9\) ft (the sum of the left triangle's base, the rectangle's width, and the right triangle's base), and the bottom base of the trapezoid is the rectangle's width (\(4\) ft), and the height of the trapezoid is \(3\) ft. Then the rectangle is \(4\) ft by \(8\) ft. So let's recalculate:

Area of trapezoid: \(\frac{(4 + 9)}{2}\times3=\frac{13}{2}\times3 = 19.5\)

Area of rectangle: \(4\times8 = 32\)

Total area: \(19.5+32 = 51.5\) square feet.

Wait, why the difference? Because when I split into two triangles and a rectangle, I think I missed that the two triangles and the rectangle's top form a trapezoid. Let's check with the triangle method: the trapezoid area is equal to the area of the two triangles plus the area of the rectangle's top (which is a rectangle with area \(4\times3\))? Wait, no. The trapezoid with bases \(4\) and \(9\) and height \(3\): \(\frac{(4 + 9)}{2}\times3=19.5\). The two triangles: left triangle area \(\frac{1}{2}\times3\times3 = 4.5\), right triangle area \(\frac{1}{2}\times2\times3 = 3\), and the middle rectangle (top of the lower rectangle) area \(4\times3 = 12\). Then \(4.5+3 + 12=19.5\), which matches the trapezoid area. Then the lower rectangle is \(4\times8 = 32\). So total area is \(19.5+32 = 51.5\) square feet. Ah, I see! I forgot the middle rectangle on top of the lower rectangle. The lower rectangle is \(4\) ft by \(8\) ft. The top part has a middle rectangle ( \(4\) ft by \(3\) ft) and two triangles (left: \(3\) ft base, \(3\) ft height; right: \(2\) ft base, \(3\) ft height). So:

Area of middle top rectangle: \(4\times3 = 12\)

Area of left triangle: \(\frac{1}{2}\times3\times3 = 4.5\)

Area of right triangle: \(\frac{1}{2}\times2\times3 = 3\)

Area of lower rectangle: \(4\times8 = 32\)

Total area: \(12 + 4.5+3+32=51.5\,\text{ft}^2\)

Yes, that makes sense. So the correct way is to consider the top part as a middle rectangle and two triangles, or a trapezoid, and the lower rectangle.

So let's redo the steps correctly:

Step1: Identify the components

- Lower rectangle: width \(w = 4\,\text{ft}\), height \(h_1 = 8\,\text{ft}\)
- Top middle rectangle: width \(w = 4\,\text{ft}\), height \(h_2 = 3\,\text{ft}\)
- Left triangle: base \(b_1 = 3\,\text{ft}\), height \(h_2 = 3\,\text{ft}\)
- Right triangle: base \(b_2 = 2\,\text{ft}\), height \(h_2 = 3\,\text{ft}\)

Step2: Calculate the area of the lower rectangle

\(A_{\text{lower rectangle}}=4\times8 = 32\,\text{ft}^2\)

Step3: Calculate the area of the top middle rectangle

\(A_{\text{top middle rectangle}}=4\times3 = 12\,\text{ft}^2\)

Step4: Calculate the area of the left triangle

\(A_{\text{left triangle}}=\frac{1}{2}\times3\times3 = 4.5\,\text{ft}^2\)

Step5: Calculate the area of the right triangle

\(A_{\text{right triangle}}=\frac{1}{2}\times2\times3 = 3\,\text{ft}^2\)

Step6: Sum all areas

\(A_{\text{total}}=32 + 12+4.5 + 3=51.5\,\text{ft}^2\)

Answer:

\(51.5\)