Question

Frank needs to check the area enclosed by the figure. The figure is made by attaching semi circles to each side of a 48 m x 48 m². Frank says the area is 1313. 28 m². find the area enclosed by the figure. use 3.14 for pie. what error might Frank have made?

Answer 1

The correct area enclosed by the figure is 9538.56 m², not 1313.28 m² as claimed by Frank. Frank's error likely occurred when subtracting the area of the square from the area of the four semicircles.

Frank needs to determine the area enclosed by a figure created by attaching four semicircles to each side of a square with dimensions 48 m x 48 m.

1. Find the area enclosed by the figure.

2. Identify the error made by Frank, who initially claimed the area was 1313.28 m².

1. Find the area of the 4 semi-circles:

The area of one full circle is given by the formula A = πr², where r is the radius. Since these are semicircles, we need to halve the result.

`\( Area_(semicircle) = (1)/(2) \pi ((48)/(2))² \)`

Calculate this for one semicircle, and then multiply by 4 for all four.

2. Find the area of the square:

Area_{square} = 48 × 48

3. The area enclosed by the figure:

Add the area of the 4 semicircles to the area of the square.

Area_{enclosed} = Area_{square} + Area_{4 semicircles}

Error made by Frank:

Frank claimed the area was 1313.28 m², but this seems to be incorrect.

Area_{enclosed} = 4 × 1/2 × π × (48/2)² + 48 × 48

Area_{enclosed} = 4 × 1/2 × 3.14 × 24² + 48 × 48

Area_{enclosed} = 4 × 3.14 × 576 + 48 × 48

Area_{enclosed} = 7234.56 + 2304

Area_{enclosed} = 9538.56 m²

Complete question:

Frank needs to check the area enclosed by the figure. The figure is made by attaching semi circles to each side of a 48 m x 48 m². Frank says the area is 1313. 28 m². find the area enclosed by the figure. use 3.14 for pie. what error might Frank have made?

Answer 2

Area is 5921.28 m^2

Frank made an error by subtracting the area of the square from the area of the 4 semi-circles.

Step-by-step explanation

Given:

Desired outcome:

1. Area enclosed by the figure

2. The error made by Frank.

Find the area of the 4 semi-circles

`\begin{gathered} Area\text{ of a semi-circle = }\frac{\pi\text{ r}^2}{2} \\ \text{ = }(3.14*(24)^2)/(2) \\ \text{ = 904.32} \end{gathered}`
`\begin{gathered} Area\text{ of 4 semi-circles = 4}*904.32 \\ \text{ = 3617.28 m}^2 \end{gathered}`

Find the area of the square

`\begin{gathered} Area\text{ of a square = l}^2 \\ \text{ = 48}^2 \\ \text{ = 2304 m}^2 \end{gathered}`

The area enclosed by the figure

The area enclosed by the figure is determined by adding the area of the 4 semi-circles to the area of the square.

`\begin{gathered} Area\text{ enclosed by the figure = Area of the square + area of the 4 semi-circles} \\ \text{ = 2304 + 3617.28} \\ \text{ = 5921.28 m}^2 \end{gathered}`

Error made by Frank

Frank made an error by subtracting the area of the square from the area of the 4 semi-circles

i.e:

`3617.28\text{ - 2304 = 1313.28 m}^2`

Hence, the area enclosed by the figure is 5921.28 m^2 and Frank made an error by subtracting the area of the square from the area of the 4 semi-circles.