what is the area of this figure?\nif necessary, round your answer to the nearest hundredth.\nsquare…

Explanation:

Step1: Identify the composite shapes

The figure consists of a large rectangle and a quarter-circle.

Step2: Calculate the large rectangle area

The rectangle has a width of $9\text{ mm}$ and a height of $9\text{ mm}$. $$A_{rect} = 9 \times 9 = 81\text{ mm}^2$$

Step3: Determine the quarter-circle radius

The radius is the total bottom length minus the rectangle's bottom part. $$r = 11 - (9 - 5) = 11 - 4 = 7\text{ mm}$$

Step4: Calculate the quarter-circle area

Use the formula for the area of a quarter-circle. $$A_{quarter} = \frac{1}{4} \pi r^2 = \frac{1}{4} \pi (7)^2 = \frac{49\pi}{4} \approx 38.4845\text{ mm}^2$$

Step5: Subtract the missing rectangular notch

A small $2\text{ mm}$ by $(9-5)\text{ mm}$ area is missing from the bottom left. $$A_{notch} = 2 \times 4 = 8\text{ mm}^2$$

Step6: Calculate the total area

Sum the rectangle and quarter-circle, then subtract the notch and the top-right cutout. $$A_{total} = 81 + 38.4845 - 8 - (4 \times (11-7)) = 81 + 38.4845 - 8 - 16 = 95.4845\text{ mm}^2$$ Alternatively, splitting into three parts: Left rectangle $(9 \times 5)$, middle rectangle $((9-4) \times (9-5))$, and quarter circle. $$A = (9 \times 5) + (5 \times 4) + 38.4845 = 45 + 20 + 38.4845 = 103.4845\text{ mm}^2$$ Wait, re-evaluating the geometry: The figure is a $9 \times 9$ square with a $2 \times 4$ notch removed, plus a quarter circle of radius $5$ (since $9-4=5$). $$A = (81 - 8) + \frac{1}{4}\pi(5^2) = 73 + 19.6349 = 92.6349\text{ mm}^2$$ Let's re-examine the image dimensions: Top is $9$, left is $9$. Bottom left segment is $5$. Vertical notch is $2$. Bottom total is $11$. The rectangle part is $9 \times 9$ minus a $4 \times 5$ top-right chunk and a $2 \times 4$ bottom-left chunk. Area of polygon = $(9 \times 9) - (4 \times (9-5)) - (2 \times (9-5)) = 81 - 16 - 8 = 57$. Radius of quarter circle = $11 - (9-5) = 7$. Total Area = $57 + \frac{1}{4}\pi(7^2) = 57 + 38.4845 = 95.4845$. Let's try another decomposition: Left rectangle $9 \times 5 = 45$. Middle rectangle $(9-4-2) \times (9-5) = 3 \times 4 = 12$. Quarter circle $\frac{1}{4}\pi(7^2) = 38.4845$. Total = $45 + 12 + 38.4845 = 95.4845$. Rounding to the nearest hundredth: $95.48$.

Answer:

95.48 square millimeters