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 3.0 m, and the area of the patio is \\(\square\\) \\(\text{m}^2\\). the area of the sunshade is \\(\square\\) \\(\text{m}^2\\). the area not covered by the sunshade is the difference of these areas. thus, the area not covered by the sunshade is \\(\square\\) \\(\text{m}^2\\).
(type integers or decimals rounded to the nearest tenth as needed.)

Explanation:

Step1: Calculate the area of the circle (patio)

The formula for the area of a circle is \( A_{circle} = \pi r^2 \). Given the radius \( r = 3.0 \, \text{m} \), we substitute into the formula:
\( A_{circle} = \pi \times (3.0)^2 = 9\pi \approx 28.3 \, \text{m}^2 \) (rounded to the nearest tenth)

Step2: Calculate the area of the triangle (sunshade)

The triangle has three altitudes and corresponding bases. We can use the formula for the area of a triangle \( A_{triangle} = \frac{1}{2} \times base \times height \). Let's use one of the base - height pairs. For example, if we take base \( = 3.1 \, \text{m} \) and height \( = 1.8 \, \text{m} \) (from the right - angled part), or we can also use other pairs. Let's use the three - part calculation (since it's a triangle with three heights). But a simpler way: the area of a triangle can also be calculated as the sum of the areas of three right - angled triangles? Wait, no, the triangle in the circle: actually, we can use the formula \( A=\frac{1}{2}(a\times h_a + b\times h_b+ c\times h_c)/2\)? No, better to use the fact that for a triangle, the area is \( \frac{1}{2}\times \) perimeter \( \times \) in - radius? No, from the diagram, we can see that the triangle has sides with corresponding heights. Let's use the three given heights and bases. Wait, the triangle is inside the circle, and we can calculate its area by adding the areas of three right - angled triangles? Wait, no, the triangle is a single triangle. Let's use the formula \( A=\frac{1}{2}\times base_1\times height_1+\frac{1}{2}\times base_2\times height_2+\frac{1}{2}\times base_3\times height_3 \). Given the sides: let's assume the three bases are \( 3.1 \, \text{m}, 2.5 \, \text{m}, 2.4 \, \text{m} \) and corresponding heights \( 1.8 \, \text{m}, 2.4 \, \text{m}, 2.5 \, \text{m} \) (from the right angles). Then:
\( A_{triangle}=\frac{1}{2}(3.1\times1.8 + 2.5\times2.4+2.4\times2.5) \)
First, calculate each product:
\( 3.1\times1.8 = 5.58 \), \( 2.5\times2.4 = 6 \), \( 2.4\times2.5 = 6 \)
Then, \( A_{triangle}=\frac{1}{2}(5.58 + 6+6)=\frac{1}{2}(17.58)=8.79\approx 8.8 \, \text{m}^2 \) (rounded to the nearest tenth)

Step3: Calculate the area not covered

The area not covered is the difference between the area of the circle and the area of the triangle. So \( A = A_{circle}-A_{triangle} \)
\( A=28.3 - 8.8 = 19.5 \, \text{m}^2 \) (Wait, maybe my calculation of the triangle area is wrong. Let's recalculate the triangle area. Another way: the triangle is an isoceles or scalene? Wait, the sum of the areas of the three right - angled triangles: actually, the triangle in the circle, the area of the triangle can be calculated as \( \frac{1}{2}\times(3.1\times1.8 + 2.5\times2.4+2.4\times2.5) \). Wait, \( 3.1\times1.8 = 5.58 \), \( 2.5\times2.4 = 6 \), \( 2.4\times2.5 = 6 \). Sum is \( 5.58 + 6+6 = 17.58 \), half of that is \( 8.79 \approx 8.8 \). The area of the circle is \( \pi\times3^2=9\pi\approx28.2743 \approx 28.3 \). Then the difference is \( 28.3 - 8.8 = 19.5 \)? Wait, maybe I made a mistake in the triangle area. Let's use another approach. The triangle has a perimeter? No, the correct way: the area of the triangle is the sum of the areas of three triangles with the center of the circle as the common vertex. So each of the three sub - triangles has an area of \( \frac{1}{2}\times side\times height \), where the height is the distance from the center to the side (the radius segments). Wait, actually, the area of the triangle is \( \frac{1}{2}(a\times r_a + b\times r_b + c\times r_c) \), where \( a,b,c \) are the sides of the triangle and \( r_a,r_b,r_c \) are the distances from the center to the sides (the apothems). Given \( a = 3.1 \, r_a = 1.8 \); \( b = 2.5 \, r_b = 2.4 \); \( c = 2.4 \, r_c = 2.5 \). Then \( A=\frac{1}{2}(3.1\times1.8 + 2.5\times2.4+2.4\times2.5)=\frac{1}{2}(5.58 + 6+6)=\frac{1}{2}(17.58) = 8.79\approx8.8 \). The area of the circle is \( \pi r^2=\pi\times3^2 = 9\pi\approx28.27 \). Then the area not covered is \( 28.27-8.79 = 19.48\approx19.5 \, \text{m}^2 \) (rounded to the nearest tenth)

Answer:

The area of the patio (circle) is approximately \( 28.3 \, \text{m}^2 \), the area of the sunshade (triangle) is approximately \( 8.8 \, \text{m}^2 \), and the area not covered by the sunshade is approximately \( \boldsymbol{19.5} \, \text{m}^2 \) (the values may vary slightly depending on the rounding method, but the main steps are to calculate the area of the circle and the area of the triangle and then find their difference).