Difference between revisions of "1971 AHSME Problems/Problem 15"
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== Solution == | == Solution == | ||
− | <math>\boxed{\textbf{(B) }3"}</math>. | + | Let the third dimension of the aquarium be <math>x</math>, and let the height of the water when the aquarium is level be <math>h</math>. When the aquarium is tilted, the water forms a triangular prism. The triangular faces have height <math>8</math> and, from the problem, base <math>\tfrac34x</math>. Thus, they have area <math>\tfrac12\cdot8\cdot\tfrac34x=3x</math>, so, because the prism has length <math>10</math>, the volume of the water is <math>10\cdot3x=30x</math>. When the tank is level, the water forms a rectangular prism with volume <math>h\cdot10\cdot x=10hx</math>. Because the amount of water in the tank is conserved, we can equate these two expressions for the volume of the water. Thus, <math>10hx=30x</math>, so <math>h=\boxed{\textbf{(B) }3"}</math>. |
== See Also == | == See Also == | ||
{{AHSME 35p box|year=1971|num-b=14|num-a=16}} | {{AHSME 35p box|year=1971|num-b=14|num-a=16}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 10:25, 9 August 2024
Problem
An aquarium on a level table has rectangular faces and is inches wide and inches high. When it was tilted, the water in it covered an end but only of the rectangular bottom. The depth of the water when the bottom was again made level, was
Solution
Let the third dimension of the aquarium be , and let the height of the water when the aquarium is level be . When the aquarium is tilted, the water forms a triangular prism. The triangular faces have height and, from the problem, base . Thus, they have area , so, because the prism has length , the volume of the water is . When the tank is level, the water forms a rectangular prism with volume . Because the amount of water in the tank is conserved, we can equate these two expressions for the volume of the water. Thus, , so .
See Also
1971 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 | ||
All AHSME Problems and Solutions |
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