Question

find the approximate area of the shaded region below, consisting of a right triangle with a circle cut out of it. use 3.14 as an approximation for π. the triangle has two sides of 56 m, and the circle has a diameter (or radius? the text in the circle is unclear, but likely related to radius/diameter for area calculation). the options are: 314 square meters, 1,254 square meters, 2,822 square meters, 312 square meters.

Explanation:

Step1: Calculate area of triangle

The triangle is a right triangle with legs \( a = 56\,\text{m} \) and \( b = 56\,\text{m} \). The area of a right triangle is \( A_{\text{triangle}}=\frac{1}{2}ab \). So, \( A_{\text{triangle}}=\frac{1}{2}\times56\times56 = 1568\,\text{m}^2 \). Wait, no, wait—wait, the circle has a diameter? Wait, the circle's diameter? Wait, the problem says the circle is cut out. Wait, the circle's radius? Wait, the circle in the triangle: the diameter? Wait, the options: let's re - check. Wait, maybe the triangle is isoceles right triangle with legs 56m. Then area of triangle is \( \frac{1}{2}\times56\times56=1568 \)? But the options are 314, 1254, 2822, 312. Wait, maybe I misread the triangle. Wait, maybe the triangle is a right triangle with legs 56m, and the circle has a diameter of 20m? Wait, the circle has "20 m" (diameter? Or radius?). Wait, the problem says "Use 3.14 as an approximation for \( \pi \)". Let's assume the circle has a diameter of 20m, so radius \( r = 10\,\text{m} \). Then area of circle is \( A_{\text{circle}}=\pi r^{2}=3.14\times10^{2}=314\,\text{m}^2 \). Then area of triangle: wait, maybe the triangle is not 56x56. Wait, no, maybe the triangle is a right triangle with legs 56m, but the circle is inside. Wait, the options: let's recalculate. Wait, maybe the triangle's area is \( \frac{1}{2}\times56\times56 = 1568 \), then subtract the circle's area. If the circle has radius 10 (diameter 20), area is 314. Then \( 1568 - 314=1254 \)? No, 1568 - 314 = 1254? Wait, 1568 - 314 = 1254? Wait, 1568 - 300 = 1268, minus 14 is 1254. So \( A_{\text{shaded}}=A_{\text{triangle}}-A_{\text{circle}} \). So \( A_{\text{triangle}}=\frac{1}{2}\times56\times56 = 1568 \), \( A_{\text{circle}} = 3.14\times10^{2}=314 \) (if radius is 10, diameter 20). Then \( 1568 - 314 = 1254 \). Wait, but the options have 1,254 square meters. Wait, maybe the triangle is not 56x56. Wait, maybe the triangle has legs of length 60? No, the diagram shows 56m. Wait, maybe I made a mistake. Wait, let's check the options. The options are 314, 1254, 2822, 312. So 1568 - 314 = 1254. So that's one of the options.

Step2: Calculate area of circle

The circle's radius \( r = 10\,\text{m} \) (assuming diameter is 20m, as the circle is labeled with 20m? Wait, the circle has "20 m"—maybe diameter. So radius \( r=\frac{20}{2}=10\,\text{m} \). Area of circle \( A_{\text{circle}}=\pi r^{2}=3.14\times10^{2}=314\,\text{m}^2 \).

Step3: Calculate shaded area

Shaded area is area of triangle minus area of circle. \( A_{\text{shaded}}=\frac{1}{2}\times56\times56 - 314 \). \( \frac{1}{2}\times56\times56=1568 \), \( 1568 - 314 = 1254\,\text{m}^2 \).

Answer:

1,254 square meters