Question

The lateral surface area of a solid rectangular box is 520 cm2

and its length , breadth

and height

are in the ratio 9:4:5. Find the total surface area of the box.​

Answer 1

$\large\sf\underline{Correct\:question\::}$

The lateral surface area of a solid rectangular box is 520$\sf\:cm^{2}$ and its length , breadth and height are in the ratio 9:4:5. Find the total surface area of the box.

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$\large\sf\underline{Understanding\:the\:question\::}$

Here in this question we are given the lateral surface area ( LSA ) of a solid rectangular box as 520$\sf\:cm^{2}$ . We are also given the ratio of length, breadth and height as 9 : 4 : 5 . Using all these we need to calculate the total surface area ( TSA ) of that rectangular box . In order to solve this problem we need to first calculate the actual value of length, breadth and height , which can be done by using the formula for LSA and the given ratio . Then using the formula of TSA we can easily calculate TSA. So let's begin !

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$\large\sf\underline{Given\::}$

• Lateral surface area ( LSA ) = $\sf\:520\:cm^{2}$

• Ratio of length, breadth and height = 9 : 4 : 5

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$\large\sf\underline{To\:find\::}$

• Total surface area ( TSA ) of the box = ??

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$\large\sf\underline{Assumption\::}$

Let's assume that :

• Length be 9a

• Breadth be 4a

• Height be 5a

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$\large\sf\underline{Solution\::}$

We know that ,

$\large\fbox\red{Lateral\:surface\:area\:=\:2h(l+b)}$

• Let's substitute the given value of LSA and assumed value of l, b and h in the formula

$\sf\implies\:520\:=2 \times 5a(9a+4a)$

$\sf\implies\:520\:=2 \times 5a \times 13a$

$\sf\implies\:520\:=2 \times 65a^{2}$

$\sf\implies\:520\:=130a^{2}$

$\sf\:oR\:130a^{2}=\:520$

$\sf\implies\:a^{2}=\frac{520}{130}$

$\sf\implies\:a^{2}=\cancel{\frac{520}{130}}$

$\sf\implies\:a^{2}= 4$

$\sf\implies\:a= \sqrt{4}$

$\large{\mathfrak\purple{\implies\:a=2\:cm}}$

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Now let's substitute the value of a in the assumed value of l, b and h :

◎ Length = 9a = $\sf\:9 \times 2$ = $\small{\underline{\boxed{\mathrm\pink{18\:cm}}}}$

◎ Breadth = 4a = $\sf\:4 \times 2$ = $\small{\underline{\boxed{\mathrm\pink{8\:cm}}}}$

◎ Height = 5a = $\sf\:5 \times 2$ = $\small{\underline{\boxed{\mathrm\pink{10\:cm}}}}$

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Now we know ,

$\large\fbox\red{Total\:surface\:area\:=\:2(lb+bh+lh)}$

• Let's substitute the value of l, b and h in the formula

$\sf\implies\:TSA\:=\:2[(18 \times 8) + (8 \times 10) + (18 \times 10)]$

$\sf\implies\:TSA\:=\:2[144+ 80 + 180]$

$\sf\implies\:TSA\:=\:2[144 + 260]$

$\sf\implies\:TSA\:=\:2[404]$

$\sf\implies\:TSA\:=\:2 \times 404$

$\large{\mathfrak\purple{\implies\:TSA\:=\:808\:cm^{2}}}$

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‎┌────── ⋆⋅✦⋅⋆ ──────┐

⠀⠀⠀‎⠀⠀${\sf{{\green{Final\:AnsWers:}}}}$

• Length = 18cm

• Breadth = 8cm

• Height = 10cm

• TSA = $\sf\:808\:cm^{2}$

└────── ⋆⋅✦⋅⋆ ──────┘

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!! Hope it helps !!