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take a two-digit number. write it backwards. add it to the original number. if what you get is a perfect square, what could be the original number? find all answers

Take a two-digit number. The possible numbers for original number condition are 29, 38, 47, 56, 65, 74, 83, 92.

SOLUTON:

Given, Take a two-digit number. Write it backwards. Add it to the original number. If what you get is a perfect square,

We have to find what could be the original number? Find all answers

Now, let the original number be xy

Then, xy + yx is a perfect square.

⇒ (10x + y) + (10y + x) is perfect square

[ because we can write 21 as 2 x 10 + 1 ]

⇒ (11x + 11y) is a perfect square

⇒ 11(x + y) is perfect square.

11(x + y) to be perfect square x + y should also be 11.

Then, x + y = 11.

Now, possible values of x and y are (2, 9), (3, 8), (4, 7), (5, 6), (6, 5), (7, 4), (8, 3), (9, 1)

(10, 1) is not possible because x range is from 0 – 9 as we took it as digit in xy

Now, the originals numbers xy are 29, 38, 47, 56, 65, 74, 83, 92.

Hence, the possible numbers for original number condition are 29, 38, 47, 56, 65, 74, 83, 92.

step-by-step explanation:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

step-by-step explanation:

given that megan owns a clothing boutique.

the profits of 2010, 11, are recorded and shown as follows:

years since     2010 0     1     2     3       4       5

profits made in 1000's   39       84     111     120   111     84

we find that profit gradually increases form 2010 to 2013 upto 120,   then starts decreasing.

the profits for 2012 and 2014 are the same while that of 2011 and 2015 are also the same.

this shows that there is symmetry about the line profit = 120000

hence answer is:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

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