Question

A rectangular box has a volume of 24 cubic units. Find the new volume of the box after reducing its length to one-half and tripling its height.
A.36 cubic units
B.20 cubic units
C.42 cubic units
D.30 cubic units

Answer 1

Hint: Use the formula for the volume of rectangular box, \[V = l \times b \times h\] and solve the question further by taking new length as one-half of the original length and new height as three times the original height.

Complete step-by-step answer:
The volume of a rectangular box \[V = l \times b \times h\] . (l= length, B = breadth and h = height) …(1)
The volume of given rectangular box is 24 cubic units i.e. V= 24.
Now, we reduce the length of the rectangular box to one-half.
So, the new length of rectangular box \[L = \dfrac{l}{2}\] .
Also, we increase the height of the rectangular box to three times of the original height.
So, the new height of the rectangular box \[H = 3h\] .
There is no change in the breath of the rectangular box.
Now, we calculate the new volume of the rectangular box \[{V_N} = L \times b \times H\] . …(2)

Substitute \[L = \dfrac{l}{2}\] and \[H = 3h\] in equation (2).
 \[{V_N} = \dfrac{l}{2} \times b \times 3h\]
 \[\therefore {V_N} = \dfrac{3}{2}\left( {l \times b \times h} \right)\]
From equation (1), substitute \[V = l \times b \times h\]
 \[\therefore {V_N} = \dfrac{3}{2}V\]
Also, V= 24
 \[\therefore {V_N} = \dfrac{3}{2} \times 24\]
 \[\therefore {V_N} = 36\]
Thus, the new volume of the rectangular box will be 36 cubic units.


Note: Short-cut method:
From the formula for volume of rectangular box, it can be seen that volume is directly proportional to length and height.
Val and Vah
So, on reducing length to one-half, the volume also reduces to one-half i.e. \[\dfrac{V}{2}\] .
Now, on making height three times the original height, the new volume becomes three times i.e. \[\dfrac{{3V}}{2}\] .
Thus, the new volume of the rectangular box becomes 36 cubic units.