The volume of a rectangular parallelopiped/cuboid of a square base is 144 CC and its diagonal of the base is 6 cm. Find the height of the parallelepiped/cuboid.
Answers
Answer:
Step-by-step explanation:
Assuming that the surface is the outer surface and the diagonal is top left front corner to right rear lower corner and the sides are a, b and c, then we have the following equations.
[math]abc = 144[/math]
[math]2ab + 2bc [/math][math]= 192[/math] or [math]2b(a + c) [/math][math]= 96[/math]
[math]\sqrt{a^2 + b^2 + c^2} = 13 [/math] or [math] a^2 + b^2 +c^2 = 169[/math]
These three equations can be solved simultaneously to find the answers, but it is also possible to use a spot of number analysis to break the answer out the above.
The factors of 144 are [math]2^4 * 3^2[/math] so the three factors must use all of them between them.
All dimensions must be less than 13, the internal diagonal, so list the numbers from 1 to 12 that can only be formed with these factors. They are:
1, 2, 3, 4, 6, 8, 9 and 12. The sides can only be of these possible lengths.
Looking at equation 3 the sums of the squares add to an odd number. One of the numbers must be a 3 or a 9 for the square to be odd and the total odd, the other two are even. It is not possible to have three odd numbers. Were this odd number to be a 9 then the squares of the other two numbers have to add up to 169–81 = 88. After some study it is not possible to sum the squares of two of the above numbers to 88, so one of the sides is 3. So let's say b = 3.
Having found that the other two side multiply to 144/3 = 48. Out of the above list there are only two pairs of candidates 4 * 12 and 6 * 8.
Looking at equation 2, then 6(a + b) equals 96. This rules out the 6 and 8.
Therefore, the three sides are 3, 4, and 12
Answer:
Step-by-step explanation