1985 AHSME Problems/Problem 25
Contents
Problem
The volume of a certain rectangular solid is , its total surface area is , and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is
Solution 1
Let the side lengths be . Thus, the volume is , so and the side lengths can be written as .
The surface area is
Both values of give the same side length, the only difference is that one makes them count up and one makes them count down. We pick . (The solution proceeds the same had we picked ). Thus, the side lengths are
We have the sum of the distinct side lengths is , and since each side length repeats times, the total sum is .
Solution 2
We see let the side lengths be , , and because they form an arithmetic progression. Therefore, we have that . Therefore, . The next piece of information tells us that Dividing both sides by , which is clearly nonzero, as it is a side length, we see that . For finding the sum of the side lengths, we will need to obtain though, and so the answer is , .
See Also
1985 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Problem 26 | |
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