Final answer:
Upon solving the equation derived from the given conditions, we find that the relation between m, n, and p, given the volume of the two boxes, should be m + n + p = -2. However, this result does not match any of the answer choices provided.
Step-by-step explanation:
The student has asked to find the relation between the dimensions m, n, and p of two rectangular boxes where one box has half the volume of the other larger box with dimensions increased by 2 in each direction (m+2, n+2, p+2). To solve this, let's denote the volume of the smaller box as V. The volume of the larger box would then be 2V. Using the given information:
- V = m × n × p
- 2V = (m+2) × (n+2) × (p+2)
Dividing the second equation by the first gives us:
(m+2)×(n+2)×(p+2) / (m × n × p) = 2
(m+2)(n+2)(p+2) = 2mnp
Expanding the left side of the equation and simplifying gives us:
mnp + 2mn + 2mp + 2np + 4m + 4n + 4p + 8 = 2mnp
Canceling out the mnp terms on both sides and dividing everything by 2, we get:
mn + mp + np + 2m + 2n + 2p + 4 = mnp
Subtracting mn, mp, and np from both sides, then factoring out a 2 from the remaining terms, we obtain:
(mn + mp + np) - mn - mp - np = 2(m + n + p + 2)
Since the left side equals zero, we infer:
0 = 2(m + n + p + 2)
Dividing both sides by 2:
0 = m + n + p + 2
Subtracting 2 from both sides gives us the final relation:
m + n + p = -2
However, since the possible answer options do not include m + n + p = -2, and given m, n, and p are integers, we need to check for any possible errors in the question or in available answer choices. It appears that there might be a mistake since none of the provided options match our final result.