Verified Answer :
Answer: The volume of Prism A is 74 cubic feet, volume of Prism B is 222 cubic feet and the volume of Prism C is 222 cubic feet.
Step-by-step explanation: Given that three rectangular prisms have a combined volume of 518 cubic feet. Prism A has one-third the volume of Prism B and Prisms B and C have equal volume.
We are to find the volume of each of the three prisms.
Let, a, b and c represent the volumes of Prism A, Prism B and Prism C respectively.
The, according to the given information, we have
\(a+b+c=518~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\a=\dfrac{1}{3}b~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)\\\\b=c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)\)
Substituting the values of a and c from equations (ii) and (iii) in equation (i), we get
\(a+b+c=518\\\\\\\Rightarrow \dfrac{1}{3}b+b+b=518\\\\\\\Rightarrow \dfrac{1}{3}b+2b=518\\\\\\\Rightarrow \dfrac{b+6b}{3}=518\\\\\\\Rightarrow 7b=1554\\\\\\\Rightarrow b=\dfrac{1554}{7}\\\\\\\Rightarrow b=222.\)
From equation (iii), we get
\(b=c=222,\)
and from equation (ii), we get
\(a=\dfrac{1}{3}\times222\\\\\\\Rightarrow a=74.\)
Thus, the volume of Prism A is 74 cubic feet, volume of Prism B is 222 cubic feet and the volume of Prism C is 222 cubic feet.
- The volume of Prism A is 74 cubic feet.
- The volume of Prism B is 222 cubic feet.
- The volume of Prism C is 222 cubic feet.
Further explanation
Given:
Three rectangular prisms have a combined volume of 518 cubic feet.
- Prism A has one-third the volume of Prism B, and
- Prisms B and C have equal volume.
Question:
What is the volume of each prism?
The Process:
Prism A has \(\frac{1}{3}\) the volume of Prism B. From the denominator 3, let us draw a diagram representing the volume of Prisms B and C, then Prism A. Remember, both prisms have the same volume.
\(Prism B: \boxed{\circ}\boxed{\circ}\boxed{\circ}\)
\(Prism C: \boxed{\circ}\boxed{\circ}\boxed{\circ}\)
\(Prism A: \boxed{\circ}\) or 1 of 3 units.
From all the diagrams above, it appears that the total units are 3 + 3 + 1 = 7 units.
Three rectangular prisms have a combined volume of 518 cubic feet. Therefore, we can calculate the volume of one unit diagram.
\(\boxed{ \ 7 \ unit = 518 \ cubic \ feet \ }\)
\(\boxed{ \ 1 \ unit = ? \ cubic \ feet \ }\)
Then \(\boxed{ \ 1 \ unit = \frac{518}{7} \ cubic \ feet \ }\)
Hence, \(\boxed{\boxed{ \ 1 \ unit = 74 \ cubic \ feet \ }}\)
And now, let us calculate the volume of each prism.
The volume of Prism A: \(1 \ unit = \boxed{ \ 74 \ cubic \ feet \ }\)
The volume of Prism B: \(3 \ unit = 3 \times 74 = \boxed{ \ 222 \ cubic \ feet \ }\)
The volume of Prism C: \(3 \ unit = 3 \times 74 = \boxed{ \ 222 \ cubic \ feet \ }\)
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