Question

The dimensions of a cuboid are in a ratio 3:2:1 and the total surface area is 1078 cm2. Find the length of its one diagonal​

Answer 1

Answer:

Length of its one diagonal 7√14cm

Explanation:

In a cuboid,

Ratio of its dimensions is 3 : 2 : 1

It's T. S. A.(Total surface area) is 1078cm²

Let's assume,

$\bullet \: \rm \: length = 3x$

$\bullet \: \rm \: breadth = 2x$

$\bullet \: \rm \: height = 1x$

We know,

$\rm \bigstar \: T. S. A. \: of \: a \: cylinder = 2(lb + bh + hl)$

Where, ‘l’ denotes the length, ‘b’ is the breadth and ‘h’ height.

According to the question,

$\rm \implies \: 1078 = 2[(3x)(2x) + (2x)(1x) + (1x)(3x)]$

$\rm \implies \: 1078 = 2[6{x}^{2} + {2x}^{2} + {3x}^{2} ]$

$\rm \implies \: 1078 = 2[11{x}^{2} ]$

$\rm \implies \: 1078 = {22x}^{2}$

$\rm \implies \: \dfrac{1078}{22} = {x}^{2}$

$\rm \implies \: 49 = {x}^{2}$

$\rm \implies \: \sqrt{49} = {x}$

$\rm \implies {x} = 7$

Hence, the dimensions are :-

$\rm \bullet \: length = 3x = 3(7) = 21cm$

$\rm \bullet \: breadth = 2x = 2(7) = 14cm$

$\rm \bullet \: height = 1x = 7cm$

Now,

Diagonal of a cuboid is given by,

$\rm = \sqrt{{(l)}^{2} + {(b)}^{2} + {(h)}^{2} }$

Where, ‘l’ denotes the length, ‘b’ is breadth and ‘h’ is the height.

$\rm = \sqrt{{(21)}^{2} + {(14)}^{2} + {(7)}^{2} }$

$\rm = \sqrt{441 + 196 + 49}$

$= \sqrt{686}$

$\rm = 7\sqrt{14} cm$

${ \underline{ \rm { \therefore{Diagonal \: of \: the \:cuboid = 7 \sqrt{14}cm }}}}$