Curious Cuboid

Let the edge lengths of the cuboid be a,b,c
Given:ab=1024.....(1)
bc=81....(2)
ca=49.....(3)
Multiplying these equations we get: $(abc)^2=1024\times81\times 49=2016^2$ $\Rightarrow abc=\large {2016}$
And since the volume V=abc , $\Large\boxed{2016}$ is the required answer.

let $L,W$ and $H$ be the length, width and height of the rectangular prism respectively

then, we have

$HL=81$ $\color{#D61F06}(1)$

$HW=49$ $\color{#D61F06}(2)$

$LW=1024$ $\color{#D61F06}(3)$

from $\color{#D61F06}(1)$ , $H=\dfrac{81}{L}$ , substitute this in $\color{#D61F06}(2)$ , we have

$\dfrac{81}{L}W=49$ or $W=\dfrac{49}{81}L$ , substitute this in $\color{#D61F06}(3)$ , we have

$L^2 \left(\dfrac{49}{81}\right)=1024$ $\color{#3D99F6}\large\implies$ $L=\dfrac{288}{7}$

solving for $H$ , we get

$H=\dfrac{81}{\frac{288}{7}}=\dfrac{63}{32}$

solving for $W$ , we get

$W=\dfrac{49}{81} \times \dfrac{288}{7}=\dfrac{224}{9}$

Finally, the volume is

$V=LWH=\dfrac{288}{7} \times \dfrac{224}{9} \times \dfrac{63}{32}=$ $\boxed{\large2016}$

Let side of the cuboid be x,y,z given Xy=1024... (1) Yz=81.... (2) Za=49...(3) Multiply eq. 1,2,3... We get (XYZ)^2=(32X9X7)^2 XYZ=2016 Hence volume of cuboid is 2016

Let $x$ be the height, $y$ be the length and $z$ be the width of the cuboid. Then we have, $zy=1024,xy=81$ and $xz=49$ . We know that the volume is $xyz$ , so we have

$(zy)(xy)(xz)=1024(81)(49)$

$z^2y^2x^2=4064256$

$\sqrt{z^2y^2x^2}=\sqrt{4064256}$

$xyz=\boxed{2016~\text{cubic units}}$

If you label the picture with L=length, W=width, and H=height, then the volume of a cuboid is V= L W H. The surface area of A = 1024 = L W (according to my diagram). The surface area of B = 81 = W H and the surface area of C = 49 = L H. Substituting 1024 for L W in the volume equation yields V = 1024*H.

If we compare the surface area equations of B and C by dividing the two, you have

81/49 = (W H)/(L H) = W/L (9/7)^2 L = W

Plugging W in terms of L back into the surface area of A equation, you get L (9/7)^2 L = 1024 Solving for L gives L = 224/9

Plugging back into the surface area of A equation will give you W, too. Solving for W gives W = 288/7.

You can use the surface area equation of B or C for the rest of the problem.

81/W = H, so 81/(288/7) = 63/32 = H

Volume of Cuboid = L W H = (224/9)(288/7)(63/32) = 2016

Given A= 1024. B =81, C=49 all the above are squares of 32, 9 and 7 respectively and in the cuboid is having the square surfaces . so the volume of the cuboid is l x b x h= 32 x 9 x 7 = 2016

$\text{Volume}=\sqrt{1024^2+81^2+49^2}=\boxed{\text{2016 cubic units}}$

given :- ab = 1024 ------(1) bc = 81 ----------(2) ac = 49 ----------(3)

Multiplying these equations :- (1) (2) (3)

(abc)^2 = 1024 * 81 * 49

abc = square root of 1024 * 81 * 49

Volume = 2016