A suitcase with measures 80 cm ×
48 cm × 24 cm is to be covered with
a tarpaulin cloth. How many metres of tarpaulin of width 96 cm is required to cover
100 such suitcases?
Please give a detailed explanation
free points
Answers
Given :
Ratio of sides of the triangle = 5 : 4 : 3
Perimeter of thetriangle = 36 cm
To Find :
The area of the triangle using heron's formula
Solution :
Let the ratio constant be " x " . Then sides of the triangle are 5x , 4x and 3x.
Perimeter of a triangle is given by ,
\begin{gathered} \\ \star \: {\boxed{\purple{\sf{Perimeter_{(triangle)} = a + b + c}}}} \\ \\ \end{gathered}
⋆
Perimeter
(triangle)
=a+b+c
Here ,
a , b and c are sides of triangle
Substituting the values we have in the formula ,
\begin{gathered} \\ : \implies \sf \: 36 = 5x + 4x + 3x \\ \\ \end{gathered}
:⟹36=5x+4x+3x
\begin{gathered} \\ : \implies \sf \: 36 = 12x \\ \\ \end{gathered}
:⟹36=12x
\begin{gathered} \\ : \implies \sf \: x = \frac{36}{12} \\ \\ \end{gathered}
:⟹x=
12
36
\begin{gathered} \\ : \implies{\underline{\boxed{\blue{\mathfrak{x = 3}}}}} \\ \\ \end{gathered}
:⟹
x=3
Then , Sides of the given triangle are ;
5x = 5(3) = 15 cm
4x = 4(3) = 12 cm
3x = 3(3) = 9 cm
\qquad ━━━━━━━━━━━━━━━━━
Now , Area of triangle (heron's formula) is given by ;
\begin{gathered} \\ \star \: {\boxed{\purple{\sf{Area_{(triangle)} = \sqrt{s(s - a)(s - b)(s - c)} }}}} \\ \\ \end{gathered}
⋆
Area
(triangle)
=
s(s−a)(s−b)(s−c)
Here ,
s is semiperimeter
a , b and c are sides of triangle
Now , Calculating the semiperimeter of given triangle ;
\begin{gathered} \\ : \implies \sf \: s = \frac{15+9+12}{2} \\ \\ \end{gathered}
:⟹s=
2
15+9+12
\begin{gathered} \\ : \implies \sf \: s = \frac{36}{2} \\ \\ \end{gathered}
:⟹s=
2
36
\begin{gathered} \\ : \implies{\underline{\boxed {\blue{\mathfrak{s = 18 \: cm}}}}} \\ \\ \end{gathered}
:⟹
s=18cm
Substituting the values we have in the heron's formula ;
\begin{gathered} \\ : \implies \sf \: Area_{(triangle)} = \sqrt{18(18- 15)(18 - 9)(18 - 12)} \\ \\ \end{gathered}
:⟹Area
(triangle)
=
18(18−15)(18−9)(18−12)
\begin{gathered} \\ : \implies \sf \: Area_{(triangle)} = \sqrt{18(3)(9)(6)} \\ \\ \end{gathered}
:⟹Area
(triangle)
=
18(3)(9)(6)
\begin{gathered} \\ : \implies \sf \: Area_{(triangle)} = \sqrt{2916} \\ \\ \end{gathered}
:⟹Area
(triangle)
=
2916
\begin{gathered} \\ : \implies{\underline{\boxed{\pink{\mathfrak{Area_{(triangle)} = 54 \: {cm}^{2} }}}}} \: \bigstar \\ \\ \end{gathered}
:⟹
Area
(triangle)
=54cm
2
★
Hence ,
The Area of the given triangle is 54 cm²
Answer:
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