Question

what is the area of this figure? provide an answer accurate to the nearest tenth. use 3.14 or a calculator button for pi if needed. 33 cm 18 cm 15 cm 21 cm 9 cm 12 cm .label required

Explanation:

Step1: Analyze the figure's components

The figure can be considered as a combination of a rectangle, a smaller rectangle (the indent), and a quarter - circle (the curved part). First, let's find the area of the large rectangle without considering the indent and the curve. The large rectangle has a length of 33 cm and a height of 21 cm. The area of a rectangle is given by \(A = l\times w\), so the area of the large rectangle is \(33\times21=693\) \(cm^{2}\).

Step2: Calculate the area of the indent (smaller rectangle)

The indent is a rectangle with length 15 cm and height 9 cm. Using the rectangle area formula, its area is \(15\times9 = 135\) \(cm^{2}\). But wait, there is also a curved part. The curved part is a quarter - circle. Let's find the radius of the quarter - circle. The vertical side of the indent is 9 cm, and the left side of the figure has a height of 18 cm, and the total height is 21 cm. Wait, actually, the radius of the quarter - circle: the difference in height between 21 cm and 18 cm? No, looking at the horizontal and vertical dimensions. The horizontal length of the curved part's "missing" rectangle: the width of the curved part's base. Wait, the length of the part where the curve is: the horizontal length from the left - most point to the start of the indent. The total length is 33 cm, the indent's length is 15 cm, and the right - most rectangle has a length of 12 cm. So \(33-(15 + 12)=6\) cm. And the vertical length: \(21 - 18=3\) cm? No, that's not right. Wait, the curved part is a quarter - circle. Let's see, the height of the curved part's "rectangle" is \(21 - 18 = 3\) cm? No, maybe the radius of the quarter - circle is \(r=21 - 18=3\) cm? Wait, no. Wait, the vertical side of the indent is 9 cm, and the left side of the figure has a height of 18 cm, and the total height is 21 cm. So the vertical distance from the bottom of the curved part to the top of the indent is \(21 - 18=3\) cm? No, maybe the radius is \(r = 21-18 = 3\) cm? Wait, actually, the curved part is a quarter - circle, so the area of a quarter - circle is \(\frac{1}{4}\pi r^{2}\). Let's find the radius. The horizontal length of the curved part's base: the length from the left - most edge to the start of the indent's top. The total length is 33 cm, the indent is 15 cm, and the right rectangle is 12 cm, so \(33-(15 + 12)=6\) cm. The vertical length from the bottom of the curved part to the top of the indent: \(21 - 18 = 3\) cm? No, that's not. Wait, maybe the radius is \(r=21 - 18 = 3\) cm? Wait, no, let's re - examine. The figure: the left - hand side has a height of 18 cm, the total height is 21 cm, so the vertical segment from the bottom of the curved part to the top of the 18 cm height is \(21 - 18 = 3\) cm. And the horizontal segment from the left - most edge to the start of the indent's top is \(33-(15 + 12)=6\) cm. Wait, but the curved part is a quarter - circle, so if the horizontal and vertical segments are equal (since it's a quarter - circle, the radius should be equal to the length of the missing square's side). Wait, maybe the radius \(r = 6\) cm? No, that doesn't fit. Wait, maybe I made a mistake in decomposing the figure. Let's try another approach.

Alternative approach: The figure can be seen as:

1. A large rectangle with length 33 cm and height 21 cm: area \(A_1=33\times21 = 693\) \(cm^{2}\)

2. Minus the area of the indent (a rectangle with length 15 cm and height 9 cm): \(A_2 = 15\times9=135\) \(cm^{2}\)

3. Plus the area of the quarter - circle (because the curved part is a quarter - circle that replaces a square - like indent). Wait, no, actually, the curved part is a quarter - circle, so we need to find the area of the quarter - circle and adjust.

Wait, the vertical side of the indent is 9 cm, and the left side of the figure has a height of 18 cm, so the vertical distance from the bottom of the curved part to the top of the 18 cm height is \(21 - 18 = 3\) cm? No, that's not. Wait, the curved part is a quarter - circle. Let's look at the horizontal and vertical dimensions. The horizontal length of the curved part's "base" is \(33-(15 + 12)=6\) cm, and the vertical length is \(21 - 18 = 3\) cm? No, that can't be. Wait, maybe the radius of the quarter - circle is \(r = 21 - 18=3\) cm? No, that seems too small. Wait, maybe the correct way is:

The figure has a rectangle of length 33 cm and height 21 cm. Then, we have a rectangular indent of length 15 cm and height 9 cm, and a quarter - circular indent (since the curve is a quarter - circle). Wait, no, the curve is a quarter - circle, so the area of the quarter - circle is added or subtracted? Let's think again.

The left - hand side of the figure: from the bottom, the height is 18 cm, and the total height is 21 cm, so the vertical segment above 18 cm is \(21 - 18 = 3\) cm. The horizontal segment to the left of the indent: the length from the left - most edge to the start of the indent's top is \(33-(15 + 12)=6\) cm. Wait, 6 cm and 3 cm? No, maybe the radius of the quarter - circle is \(r = 6\) cm? Wait, no, the curved part is a quarter - circle, so the area of the quarter - circle is \(\frac{1}{4}\pi r^{2}\). Let's assume that the radius \(r = 21 - 18=3\) cm? No, that's not. Wait, maybe I messed up the decomposition.

Wait, another way: The figure can be divided into:

- A rectangle with length 33 cm and height 18 cm: area \(A_{rect1}=33\times18 = 594\) \(cm^{2}\)

- A rectangle with length 12 cm and height \(21 - 18 = 3\) cm: area \(A_{rect2}=12\times3 = 36\) \(cm^{2}\)

- A quarter - circle: the radius of the quarter - circle is \(r = 21 - 18=3\) cm? No, wait, the horizontal length of the curved part's base is \(33-(15 + 12)=6\) cm, and the vertical length is \(21 - 18 = 3\) cm? No, that's not. Wait, maybe the radius is \(r = 6\) cm? Wait, no, let's calculate the area step by step.

Wait, let's find the area of the figure by adding the areas of its components:

1. The main rectangle: length 33 cm, height 21 cm: \(A_1 = 33\times21=693\)

2. The indent: a rectangle with length 15 cm, height 9 cm: \(A_2 = 15\times9 = 135\)

3. The quarter - circle: the radius of the quarter - circle. The vertical side of the indent is 9 cm, and the left side of the figure has a height of 18 cm, so the vertical distance from the bottom of the curved part to the top of the 18 cm height is \(21 - 18 = 3\) cm. The horizontal distance from the left - most edge to the start of the indent's top is \(33-(15 + 12)=6\) cm. Wait, the curved part is a quarter - circle, so the area of the quarter - circle is \(\frac{1}{4}\pi r^{2}\), where \(r\) is the radius. If we look at the right angle of the quarter - circle, the two sides of the right angle are equal to the radius. So if the horizontal and vertical segments are equal, then \(r = 6\) cm? No, 6 cm and 3 cm are not equal. Wait, maybe the correct radius is \(r = 21 - 18 = 3\) cm, and the horizontal segment is 6 cm, which is a mistake. Wait, maybe the figure has a quarter - circle with radius \(r=21 - 18 = 3\) cm, and the area of the quarter - circle is \(\frac{1}{4}\times3.14\times3^{2}=\frac{1}{4}\times3.14\times9 = 7.065\) \(cm^{2}\)

Now, the total area:

Area of large rectangle (\(33\times21\)) minus area of indent rectangle (\(15\times9\)) plus area of quarter - circle.

Wait, \(33\times21=693\), \(15\times9 = 135\), so \(693-135 = 558\). Then, the quarter - circle: if the radius is \(r = 21 - 18=3\) cm, the area of the quarter - circle is \(\frac{1}{4}\times3.14\times3^{2}=7.065\). Wait, but is the quarter - circle added or subtracted?

Wait, the original figure (without the indent and the curve) would be a rectangle. Then, we have a rectangular indent (area to subtract) and a quarter - circular "bump" (area to add) or indent? Wait, looking at the figure, the curve is a concave curve, so it's a quarter - circular indent. Wait, no, the curve is a quarter - circle, so the area of the quarter - circle is subtracted? Wait, no, let's look at the coordinates.

Alternatively, let's use the correct decomposition:

The figure can be considered as:

- A rectangle with length \(33\) cm and height \(21\) cm: \(A_1 = 33\times21=693\)

- A rectangular cut - out with length \(15\) cm and height \(9\) cm: \(A_2 = 15\times9 = 135\)

- A quarter - circular cut - out with radius \(r=21 - 18 = 3\) cm: The area of a quarter - circle is \(\frac{1}{4}\pi r^{2}=\frac{1}{4}\times3.14\times3^{2}=7.065\)

Wait, no, the quarter - circle is a part of the cut - out? Wait, the left - hand side of the cut - out (the indent) has a curved edge, which is a quarter - circle. So the area of the indent is the area of the rectangle (\(15\times9\)) minus the area of the quarter - circle? No, that doesn't make sense.

Wait, maybe the correct radius is \(r = 9-(21 - 18)=6\) cm? Wait, \(21 - 18 = 3\), \(9 - 3=6\). Ah, that makes sense. The vertical height of the indent is 9 cm, and the height from the bottom to the 18 cm mark is 18 cm, so the vertical segment within the indent is \(9-(21 - 18)=6\) cm. So the radius of the quarter - circle is \(r = 6\) cm.

Yes, that makes sense. So the radius \(r = 9-(21 - 18)=6\) cm. Then the area of the quarter - circle is \(\frac{1}{4}\pi r^{2}=\frac{1}{4}\times3.14\times6^{2}=\frac{1}{4}\times3.14\times36 = 28.26\) \(cm^{2}\)

Now, let's recalculate:

1. Area of the large rectangle: \(33\times21 = 693\) \(cm^{2}\)

2. Area of the rectangular indent: \(15\times9 = 135\) \(cm^{2}\)

3. Area of the quarter - circle (which is part of the indent, so we subtract the rectangle and add the quarter - circle, because the indent has a rectangular part and a quarter - circular part. Wait, the indent is a rectangle with a quarter - circle removed? No, the indent is a shape that has a rectangle and a quarter - circle. Wait, the figure's indent: the bottom part of the indent is a rectangle, and the left part is a quarter - circle. So the area of the indent is the area of the rectangle minus the area of the quarter - circle.

So the area of the figure is:

Area of large rectangle - (Area of rectangular indent - Area of quarter - circle)

\(=33\times21-(15\times9-\frac{1}{4}\pi r^{2})\)

We found \(r = 6\) cm (since \(9-(21 - 18)=6\))

So:

\(693-(135 - 28.26)=693 - 106.74 = 586.26\)

Wait, let's check again:

Large rectangle: \(33\times21 = 693\)

Indent: the indent is a rectangle of \(15\times9\) with a quarter - circle (area \(\frac{1}{4}\pi\times6^{2}=28.26\)) removed from its left - hand side. So the area of the indent is \(15\times9-28.26 = 135 - 28.26 = 106.74\)

Then, the area of the figure is \(693-106.74 = 586.26\), which is approximately \(586.3\) when rounded to the nearest tenth.

Wait, let's verify the radius again. The height of the indent is 9 cm, and the height from the bottom to the 18 cm mark is 18 cm, total height 21 cm, so the height above 18 cm is \(21 - 18 = 3\) cm. So the vertical length of the quarter - circle's radius is \(9 - 3=6\) cm, which matches our earlier calculation. So the radius \(r = 6\) cm.

So the steps are:

Step1: Calculate the area of the large rectangle

The large rectangle has length \(l = 33\) cm and width \(w = 21\) cm. Using the formula for the area of a rectangle \(A=l\times w\), we get \(A_1=33\times21 = 693\) \(cm^{2}\).

Step2: Calculate the area of the rectangular part of the indent

The rectangular part of the indent has length \(l = 15\) cm and width \(w = 9\) cm. Using the rectangle area formula, \(A_2=15\times9 = 135\) \(cm^{2}\).

Step3: Calculate the area of the quarter - circle

The radius of the quarter - circle \(r=9-(21 - 18)=6\) cm. The formula for the area of a circle is \(A=\pi r^{2}\), so the area of a quarter - circle is \(A_3=\frac{1}{4}\pi r^{2}\). Substituting \(r = 6\) and \(\pi = 3.14\), we get \(A_3=\frac{1}{4}\times3.14\times6^{2}=\frac{1}{4}\times3.14\times36 = 28.26\) \(cm^{2}\).

Step4: Calculate the area of the indent

The indent is a rectangle with a quarter - circle removed, so its area \(A_{indent}=A_2 - A_3=135 - 28.26 = 106.74\) \(cm^{2}\).

Step5: Calculate the area of the figure

The area of the figure is the area of the large rectangle minus the area of the indent: \(A = A_1 - A_{indent}=693 - 106.74 = 586.26\) \(cm^{2}\). Rounding to the nearest tenth, we get \(586.3\) \(cm^{2}\).

Answer:

\(586.3\) square centimeters (or \(586.3\) \(cm^{2}\))