If V and S denote the volume and Total Surface Area of a cuboid (of sides a,b,c), respectively, then which of the following is true?
- A\frac{1}{V} = \frac{2}{S} (a + b + c)
- B\frac{1}{V} = S \left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right)
- C\frac{1}{V} = \frac{2}{S} \left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right)
- DV = \frac{2}{S} \left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right)
Solution & Step-by-step Explanation
For a cuboid with side lengths a,b, and c:
The volume V is given by:
V=abc
The total surface area S is given by:
S=2(ab+bc+ca)
Let's test the expression (
a
1
+
b
1
+
c
1
):
a
1
+
b
1
+
c
1
=
abc
bc+ca+ab
We can substitute ab+bc+ca=
2
S
and abc=V into this relation:
a
1
+
b
1
+
c
1
=
V
2
S
=
2V
S
Rearranging the terms to isolate
V
1
:
V
1
=
S
2
(
a
1
+
b
1
+
c
1
)
This matches option C.
The volume V is given by:
V=abc
The total surface area S is given by:
S=2(ab+bc+ca)
Let's test the expression (
a
1
+
b
1
+
c
1
):
a
1
+
b
1
+
c
1
=
abc
bc+ca+ab
We can substitute ab+bc+ca=
2
S
and abc=V into this relation:
a
1
+
b
1
+
c
1
=
V
2
S
=
2V
S
Rearranging the terms to isolate
V
1
:
V
1
=
S
2
(
a
1
+
b
1
+
c
1
)
This matches option C.