Question

what is the area of the patio not covered by the sunshade? explain how you found your answer.
the center of the circle is the intersection of the center of the base of the triangle. so, the radius of the patio is 30 m, and the area of the patio is □ m². the area of the sunshade is □ m². the area not covered by the sunshade is the difference of these areas. thus, the area not covered by the sunshade is □ m².
(type integers or decimals rounded to the nearest tenth as needed.)

Explanation:

Step1: Find the radius of the circle (patio)

The center of the circle is the intersection of the center of the triangle. From the triangle, we can use the properties of a triangle (maybe a right triangle or using the formula for the circumradius of a triangle). Wait, looking at the triangle with sides 18m, 24m, 25m? Wait, no, the triangle has a right angle? Wait, 18 - 24 - 30? Wait, the radius is given as 30? Wait, no, the problem says "the radius of the patio is 30 m"? Wait, maybe the triangle is a right triangle with legs 18 and 24, so hypotenuse is $\sqrt{18^2 + 24^2}=\sqrt{324 + 576}=\sqrt{900}=30$. So the radius of the circle (patio) is 30 m.

Step2: Calculate the area of the patio (circle)

The formula for the area of a circle is $A = \pi r^2$. So with $r = 30$, $A_{patio}=\pi\times30^2 = 900\pi\approx 2827.43$ $m^2$.

Step3: Calculate the area of the sunshade (triangle)

The triangle is a right triangle with legs 18 m and 24 m (since $18^2 + 24^2 = 324 + 576 = 900 = 30^2$, so it's a right triangle). The area of a right triangle is $\frac{1}{2}\times base\times height$. So $A_{sunshade}=\frac{1}{2}\times18\times24 = 216$ $m^2$.

Step4: Calculate the area not covered

The area not covered is the area of the patio minus the area of the sunshade. So $A = 900\pi - 216\approx2827.43 - 216 = 2611.43$ $m^2$. Wait, but let's check again. Wait, maybe the radius is 15? No, the hypotenuse is 30, so the circumradius of a right triangle is half the hypotenuse, so $r=\frac{30}{2}=15$? Wait, that's a mistake earlier. Oh right! In a right triangle, the circumradius is half the hypotenuse. So hypotenuse is 30, so radius $r = 15$ m. Oh! That's the error. So let's correct Step1.

Step1 (corrected): Find the radius of the circle (patio)

The triangle is a right triangle with legs 18 and 24, hypotenuse 30. The circumradius of a right triangle is half the hypotenuse, so $r=\frac{30}{2}=15$ m.

Step2 (corrected): Calculate the area of the patio (circle)

$A_{patio}=\pi r^2=\pi\times15^2 = 225\pi\approx706.86$ $m^2$.

Step3 (corrected): Calculate the area of the sunshade (triangle)

Area of right triangle: $\frac{1}{2}\times18\times24 = 216$ $m^2$.

Step4 (corrected): Calculate the area not covered

$A = A_{patio}-A_{sunshade}=225\pi - 216\approx706.86 - 216 = 490.86$ $m^2$. Wait, but the problem statement said "the radius of the patio is 30 m"? Wait, the original problem text says "So, the radius of the patio is 30 m". Oh, maybe the triangle is not a right triangle? Wait, the problem text in the image says "the radius of the patio is 30 m". So let's go with that. So radius $r = 30$ m. Then area of patio is $\pi\times30^2 = 900\pi\approx2827.43$ $m^2$. Area of sunshade: if it's a triangle, maybe with base 30 and height? Wait, no, the triangle has sides 18, 24, 25? No, the labels are 18m, 24m, 25m, 31m? Wait, the image shows a triangle with a right angle, 18 and 24 as legs. So maybe the problem has a typo, but according to the problem text, the radius is 30 m. So proceeding with $r = 30$ m. Then area of patio: $900\pi\approx2827.43$, area of sunshade: 216, so area not covered: $2827.43 - 216 = 2611.43\approx2611.4$ (rounded to nearest tenth). But let's check the problem again. The problem says "the radius of the patio is 30 m", "the area of the patio is [ ]", "the area of the sunshade is [ ]", "the area not covered is the difference". So:

- Radius of patio: 30 m (given in the problem text)
- Area of patio (circle): $A = \pi r^2 = \pi\times30^2 = 900\pi \approx 2827.4$ $m^2$ (rounded to nearest tenth)
- Area of sunshade (triangle): right triangle with legs 18 and 24, so $A = \frac{1}{2}\times18\times24 = 216$ $m^2$
- Area not covered: $2827.4 - 216 = 2611.4$ $m^2$

Answer:

The area not covered by the sunshade is approximately $\boxed{2611.4}$ $m^2$.