Find the area of the composite figure.

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Burkard2020 Burkard2020

11-02-2022
Mathematics

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Find the area of the composite figure.

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MisterBrian MisterBrian · Answer 1 · 2022-02-14T17:30:53+01:00

Answer:

The Area of the composite figure would be 76.26 in^2

Step-by-step explanation:

According to the Figure Given:

Total Horizontal Distance = 14 in

Length = 6 in

To Find :

The Area of the composite figure

Solution:

Firstly we need to find the area of Rectangular part.

So We know that,

[tex]\boxed{ \rm \: Area \: of \: Rectangle = Length×Breadth}[/tex]

Here, Length is 6 in but the breadth is unknown.

To Find out the breadth, we’ll use this formula:

[tex] \boxed{\rm \: Breadth = total \: distance - Radius}[/tex]

According to the Figure, we can see one side of a rectangle and radius of the circle are common, hence,

[tex] \longrightarrow\rm \: Length \: of \: the \: circle = Radius[/tex]

Since Length = 6 in ;

[tex]\longrightarrow \rm \: 6 \: in = radius[/tex]

Hence Radius is 6 in.

So Substitute the value of Total distance and Radius:

Total Horizontal Distance= 14
Radius = 6

[tex] \longrightarrow\rm \: Breadth = 14-6[/tex]

[tex] \longrightarrow\rm \: Breadth = 8 \: in[/tex]

Hence, the Breadth is 8 in.

Then, Substitute the values of Length and Breadth in the formula of Rectangle :

Length = 6
Breadth = 8

[tex] \longrightarrow\rm \: Area \: of \: Rectangle = 6 \times 8[/tex]

[tex]\longrightarrow \rm \: Area \: of \: Rectangle = 48 \: in {}^{2} [/tex]

Then, We need to find the area of Quarter circle :

We know that,

[tex]\boxed{\rm Area_{(Quarter \; Circle) } = \cfrac{\pi{r} {}^{2} }{4}} [/tex]

Now Substitute their values:

r = radius = 6
π = 3.14

[tex]\longrightarrow\rm Area_{(Quarter \; Circle) } = \cfrac{3.14 \times 6 {}^{2} }{4} [/tex]

Solve it.

[tex]\longrightarrow\rm Area_{(Quarter \; Circle) } = \cfrac{3.14 \times 36}{4} [/tex]

[tex]\longrightarrow\rm Area_{(Quarter \; Circle) } = \cfrac{3.14 \times \cancel{{36} } \: ^{9} }{ \cancel4} [/tex]

[tex]\longrightarrow\rm Area_{(Quarter \; Circle)} =3.14 \times 9[/tex]

[tex]\longrightarrow\rm Area_{(Quarter \; Circle) } = 28.26 \: {in}^{2} [/tex]

Now we can Find out the total Area of composite figure:

We know that,

[tex]\boxed{ \rm \: Area_{(Composite Figure)} =Area_{(rectangle)}+ Area_{ (Quarter Circle)}}[/tex]

So Substitute their values:

[tex]\rm Area_{(rectangle)}[/tex] = 48
[tex]\rm Area_{(Quarter Circle)}[/tex] = 28.26

[tex]\longrightarrow \rm \: Area_{(Composite Figure)} =48 + 28 .26[/tex]

Solve it.

[tex]\longrightarrow \rm \: Area_{(Composite Figure)} =\boxed{\tt 76.26 \:\rm in {}^{2}} [/tex]

Hence, the area of the composite figure would be 76.26 in² or 76.26 sq. in.

[tex] \rule{225pt}{2pt}[/tex]

I hope this helps!