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As the problem is worded and presented, it is TERRIBLE.
Nevertheless, I was able to decipher its hidden meaning.
So, I will solve the problem, but do not ask me how I guessed my interpretation.
Let x, y and z be the three dimensions of the prism.
Then we are given these three equations
xy = 6 (1)
xz = 9 (2)
yz = 12 (3)
Multiply these equations. You will get
(xyz)^2 = 6*9*12 = 648 = (2*3) * (3*3) * (3*4) = 2 * 3^4 * 4 = 2^3 * 3^4
which implies
xyz = = = = . (4)
Now divide equation (4) by equation (1). You will get z = = .
Next, divide equation (4) by equation (2). You will get y = = .
Finally, divide equation (4) by equation (3). You will get x = = .
+-----------------------------------------------------+
| So, at this point we know all the dimensions, |
| and now I am ready answer the questions. |
+-----------------------------------------------------+
(a) To the nearest tenth of a centimeter, what is the length of the diagonal of the prism?
The square of the diagonal length is d^2 = x^2 + y^2 + z^2 = = 38.5;
hence, the diagonal is d = = 6.2 cm (rounded). ANSWER
(b) What is the volume of the prism?
The volume is V = xyz = .. = = = 25.456 cm^3 (rounded). ANSWER
The problem is solved.
All the questions are answered.
You get a brilliant solution with the careful explanation.
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