Mathematics
mathcounts questions: NEED WORK WITH ANSWERS!!!

Question Description

178. For a particular rectangular solid with integer dimensions, the sum of its length, width and height is 50 cm. What is the absolute difference between the greatest possible volume and the least possible volume of the solid?

Final Answer

OK, well the minimum volume is easy. If any of the values are zero, the volume will be zero. If "integer dimensions" is supposed to imply "positive non-zero integers", then your minimum volume is achieved by setting two sides to the minimum value of 1, and the remaining side to 48:

1 x 1 x 48 = 48cm^3

Now, there's a general rule that for any set of numbers with a fixed sum, their product is maximised if they're the same. For example, a(1-a) is max when a = (1-a) = 0.5. So the three integer values that are closest to each other, and sum to 50, are 17, 17 and 16:

17 x 17 x 16 = 4624cm^3

Which is the maximum volume for the solid (given integer dimensions).


If your question requires a proof of this, it likely involves some partial differentiation - if that's the case you can re-post under calculus specifying that - but I expect you're just required to find the logical answer.

iainharlow (1041)
Purdue University

Anonymous
Top quality work from this tutor! I’ll be back!

Anonymous
It’s my second time using SP and the work has been great back to back :) The one and only resource on the Interwebs for the work that needs to be done!

Anonymous
Thanks, good work

Studypool
4.7
Trustpilot
4.5
Sitejabber
4.4

Brown University





1271 Tutors

California Institute of Technology




2131 Tutors

Carnegie Mellon University




982 Tutors

Columbia University





1256 Tutors

Dartmouth University





2113 Tutors

Emory University





2279 Tutors

Harvard University





599 Tutors

Massachusetts Institute of Technology



2319 Tutors

New York University





1645 Tutors

Notre Dam University





1911 Tutors

Oklahoma University





2122 Tutors

Pennsylvania State University





932 Tutors

Princeton University





1211 Tutors

Stanford University





983 Tutors

University of California





1282 Tutors

Oxford University





123 Tutors

Yale University





2325 Tutors