The volume of a cuboid is 576 cm". The length of its diagonal is 244 . If the thickness of the cuboid
is 6 cm, find its length and breadth.
Answers
To Find - Length and Breadth of a Cuboid.
Solution -
Volume of Cuboid = length × breadth × thickness
576 cm^{3}cm
3
= length × breadth × 6 cm
96 cm^{2}cm
2
= length × breadth
length = \frac{96}{breadth}length=
breadth
96
Diagonal of Cuboid = \sqrt{(length)^{2} +(breadth)^{2} +(thickness)^{2} }
(length)
2
+(breadth)
2
+(thickness)
2
\sqrt{244} = \sqrt{(length)^{2} +(breadth)^{2} +(thickness)^{2} }
244
=
(length)
2
+(breadth)
2
+(thickness)
2
244 = (length)^{2} +(breadth)^{2} +(thickness)^{2}244=(length)
2
+(breadth)
2
+(thickness)
2
244 = (length)^{2} +(breadth)^{2} +36244=(length)
2
+(breadth)
2
+36
244 - 36 = (length)^{2} +(breadth)^{2}244−36=(length)
2
+(breadth)
2
208 = (length)^{2} +(breadth)^{2}208=(length)
2
+(breadth)
2
208 = (\frac{96}{breadth})^{2} +(breadth)^{2}208=(
breadth
96
)
2
+(breadth)
2
Let length = y, breadth = x, then above equation will become-
208 = \frac{9,216}{x^{2} } + x^{2}208=
x
2
9,216
+x
2
\begin{gathered}208x^{2} = 9,216 + x^{4} \\x^{4} - 208x^{2}+ 9,216 = 0\\\end{gathered}
208x
2
=9,216+x
4
x
4
−208x
2
+9,216=0
Solving the above equation by quadratic formula, we get roots as
\begin{gathered}x^{2} = 64, 144\\x = 8, -8, 12, -12\end{gathered}
x
2
=64,144
x=8,−8,12,−12
As the breadth can't be negative therefore,
x = 8, 12x=8,12
Corresponding to these values we get length as
y = 12, 8y=12,8
As length is always greater than breadth therefore only possible values are- Length = 12 cm and Breadth = 8 cm.