Two numbers are such that the square of one is 224 less than 8 times the square of the other. If
the numbers are in the ratio of 3 : 4, find the numbers.
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A.
8, 18
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B.
9, 12
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C.
12, 16
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D.
6, 8
Correct Answer:
D. 6, 8
Explanation:
To solve for the two numbers in the ratio of 3:4, let the values be represented as 3x and 4x. Based on the problem, the square of one number is 224 less than 8 times the square of the other. By testing the larger number (4x) as the one being subtracted from, we set up the equation: 8(3x)^2 - (4x)^2 = 224. Simplifying the terms gives 8(9x^2) - 16x^2 = 224, which reduces to 72x^2 - 16x^2 = 224. Subtracting the squares results in 56x^2 = 224. Dividing both sides by 56 yields x^2 = 4, meaning x = 2. Finally, multiply the ratio constants by 2 to find the numbers: 3(2) = 6 and 4(2) = 8. Therefore, the numbers are 6 and 8.
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