Question

find the skating area shown to the right. use π≈3.14. the skating area is approximately \boxed{\quad} (do not round until the final answer. then round to the nearest tenth as needed.) \text{diagram shows a rectangle with length } 23 \\, \text{m and semicircles with radius } 4 \\, \text{m on the ends.}

Explanation:

Step1: Identify the shapes

The skating area consists of a rectangle and a circle (the two semicircles on the ends make a full circle). The radius of the circle \( r = 4\) m, and the length of the rectangle \( l = 23\) m, the width of the rectangle is equal to the diameter of the circle, so \( w = 2r=8\) m.

Step2: Calculate the area of the rectangle

The area of a rectangle is given by \( A_{rectangle}=l\times w\). Substituting the values, we get \( A_{rectangle}=23\times8 = 184\) square meters.

Step3: Calculate the area of the circle

The area of a circle is given by \( A_{circle}=\pi r^{2}\). Substituting \( \pi\approx3.14\) and \( r = 4\), we get \( A_{circle}=3.14\times4^{2}=3.14\times16 = 50.24\) square meters.

Step4: Calculate the total skating area

The total area \( A = A_{rectangle}+A_{circle}\). Substituting the values, we get \( A=184 + 50.24=234.24\) square meters. Rounding to the nearest tenth, we check the hundredth place (4), which is less than 5, so we keep the tenth place as is. So \( A\approx234.2\) (wait, no, 234.24 rounded to the nearest tenth: the tenths place is 2, hundredths is 4, so it's 234.2? Wait, no, 234.24: the number is 234.24, so to the nearest tenth, look at the hundredth digit (4). Since 4 < 5, we round down, so 234.2? Wait, but let's recalculate:

Wait, rectangle length 23, width: the diameter of the circle is \( 2\times4 = 8\), so rectangle area is \( 23\times8=184\). Circle area: \( \pi r^{2}=3.14\times16 = 50.24\). Total area: \( 184 + 50.24=234.24\). Rounded to the nearest tenth is 234.2? Wait, no, 234.24: the tenths digit is 2, hundredths is 4. So when rounding to the nearest tenth, we look at the hundredth digit. If it's 5 or more, we round up the tenths digit, else we leave it. So 4 < 5, so 234.2. Wait, but maybe I made a mistake in the width. Wait, the diagram shows a rectangle with length 23 m, and on the ends, semicircles with radius 4 m. So the width of the rectangle is equal to the diameter of the semicircle, which is \( 2\times4 = 8\) m. So that part is correct. Then the two semicircles make a full circle, so area of circle is \( \pi r^{2}\). So total area is rectangle area plus circle area. So 184 + 50.24 = 234.24, which rounds to 234.2 when rounded to the nearest tenth. Wait, but maybe the problem expects the answer as 234.2 or maybe I miscalculated. Wait, let's check again:

Rectangle: length 23, width 8 (since radius is 4, diameter 8). Area: 23*8=184.

Circle: radius 4, area 3.14*16=50.24.

Total: 184+50.24=234.24. Rounded to the nearest tenth: 234.2.

Answer:

\( 234.2 \) square meters (or if we consider that maybe the width is different, but based on the diagram, radius 4, so diameter 8, so the calculation is correct. Wait, maybe the length is 23, and the width is 8, so the rectangle is 23x8, and the circle is radius 4. So the total area is 234.24, which is 234.2 when rounded to the nearest tenth.