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Question 84603This question is from textbook Linear Algebra & Its Applications
: Time to weigh the Hippopotami!
Martha is the chief hippopotamus caretaker at the Henry Vilas Zoo in Madison, Wisconsin. She has just arrived at the cargo dock in order to pick up four members of the endangered species hippopotamus mathematicus that were recently rescued from African poachers. Before the animals are released, Martha must weigh them.
However, the only scale big enough to weigh a hippo is the truck scale, which doesn't weigh anything lighter than 300 kilograms. This is more than each of the hippos weighs. Martha is puzzled for a few minutes, then gets the idea of weighing the hippos in pairs, thinking that if she gets the weight of every possible pair, she can later figure out the weights of the individual hippos.
She weighs the hippos pair by pair, and gets 312 kg, 356 kg, 378 kg, 444 kg, and 466 kg. Before she can weigh the heaviest pair of hippos, the scale breaks. Alas.
1. What was the weight of the last pair of hippos?
2. What are the weights of the individual hippos?
3. Could you have done this problem with fewer weighings? How many fewer?
This question is from textbook Linear Algebra & Its Applications
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Call the individual hippos A, B, C, and D
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There are six possible pairings for weights: (A+B), (A+C), (A+D), (B+C), (B+D), and (C+D).
Martha did all of them except (C+D).
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I solved the problem using the weights of just four of these pairs: (A+B), (A+C), (A+D),
and (B+C)
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I assumed that the pairs weigh as follows (all weights in kg and pairs in ascending
order by weight):
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A+B = 312
A+C = 356
A+D = 378
B+C = 444
B+D = 466
C+D = ???
First I added the weights of the pairs A+B and A+C as follows:
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(A+B) + (A+C) = 312 + 356 = 668
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The left side adds to give:
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2A + (B+C) = 668
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But (B+C) is a pair that was weighed at 444 kg. Substitute 444 for (B+C).
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2A + 444 = 668
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Subtracting 444 from both sides results in:
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2A = 224
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Divide both sides by 2 and you get
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A = 112
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You know that the pair A+B weighs 312. Substitute 112 for A and you get:
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112 + B = 312
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Subtract 112 from both sides and you get:
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B = 200
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Next you know that the pair A+C weighs 356. Substitute 112 for A and you get:
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112 + C = 356
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Subtract 112 from both sides and you get:
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C = 244
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Finally, you know that the pair A+D weighs 378. Substitute 112 for A and you get:
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112 + D = 378
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Subtract 112 from both sides and you get:
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D = 266
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Now answer the questions:
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1. The weight of the heaviest pair is the weight of C + D which is 244 + 266 = 510 kg
2. The weights of the individual hippos are A= 112 kg, B= 200 kg, C = 244 kg, and D = 266 kg.
3. Martha tried to find the weights of six pairs. I only needed the weights of four pairs
to solve the problem.
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Been a long time since I lived in Madison, WI, but I remember the Vilas park zoo.
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