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Elenna [48]
2 years ago
5

Prove the following statement.

Mathematics
1 answer:
gayaneshka [121]2 years ago
6 0

Answer:

You can prove this statement as follows:

Step-by-step explanation:

An odd integer is a number of the form 2k+1 where k\in \mathbb{Z}. Consider the following cases.

Case 1. If k is even we have: (2k+1)^{2}=(2(2s)+1)^{2}=(4s+1)^{2}=16s^2+8s+1=8(2s^2+s)+1.

If we denote by m=2s^2+2 we have that (2k+1)^{2}=8m+1.

Case 2. if k is odd we have: (2k+1)^{2}=(2(2s+1)+1)^{2}=(4s+3)^{2}=16s^2+24s+9=16s^{2}+24s+8+1=8(2s^{2}+3s+1)+1.

If we denote by m=2s^{2}+24s+1 we have that (2k+1)^{2}=8m+1

This result says that the remainder when we divide the square of any odd integer by 8 is 1.

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The time duration of the two pipes restricted flow before the flow became normal is 4.5 minutes

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The time duration for the volume, V, of the basin to be filled by one of the pipe, A, = 12 minutes

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Therefore, the flow rate of pipe A = V/12

The flow rate of pipe B = V/16

Due to the restriction, we have;

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The proportion of its carrying capacity the second pipe, B, carries = 5/6 of the carrying capacity

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\dfrac{7}{8} \times \dfrac{V}{12} \times t + \dfrac{5}{6} \times \dfrac{V}{16} \times t +  \dfrac{V}{12} \times 3 + \dfrac{V}{16} \times 3 = V

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