SOLUTION: The length and the width of a rectangular floor a m and b m. Solve for a and b, given the following information: • If the length and the width each increases with 1 m, the area

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Question 898081: The length and the width of a rectangular floor a m and b m. Solve for a and b, given the following
information:
• If the length and the width each increases with 1 m, the area will increase with 9 m^2.
• If the length decreases with 0.5 m and the width increases with 4 m, the area is doubled.
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
A=ab for original area.

First Part:
%28a%2B1%29%28b%2B1%29=%28A%2B9%29
%28a%2B1%29%28b%2B1%29=ab%2B9
ab%2Ba%2Bb%2Bb%5E2=ab%2B9
a%2Bb%2Bb%5E2=9

Second Part:
%28a-0.5%29%28b%2B4%29=2A
%28a-1%2F2%29%28b%2B4%29=2ab
ab-b%2F2%2B4a-2=2ab
-b%2F2%2B4a-2=ab
4a-b%2F2-2=ab
8a-b-4=2ab----- can this be solved for a in terms of b?
8a-b-4-2ab=0
8a-2ab=b%2B4
a%288-2b%29=4
a=4%2F%288-2b%29

Using First Part, solve that one for a.
a=9-b-b%5E2

Equate the formulas for a.
4%2F%288-2b%29=9-b-b%5E2
4=%288-2b%29%289-b-b%5E2%29
4=72-8b-8b%5E2-18b%2B2b%5E2%2B2b%5E3
2b%5E3-6b%5E2-26b%2B68=0
highlight_green%28b%5E3-3b%5E2-13b%2B34=0%29

From that, try rational roots theorem testing for roots of 1, 2, 17; and I doubt that 34 would be any use.

(Intentionally unfinished)

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
The length and the width of a rectangular floor a m and b m. Solve for a and b, given the following
information:
• If the length and the width each increases with 1 m, the area will increase with 9 m^2.
• If the length decreases with 0.5 m and the width increases with 4 m, the area is doubled. 


Length:  a

Width:   b

Area: ab



An increase in length, by 1 results in new length being: a + 1

An increase in width, by 1 results in new width being: b + 1

If new area increases by 9, then: (a + 1)(b + 1) = ab + 9

ab + a + b + 1 = ab + 9

ab - ab + a + b = 9 - 1

a + b = 8____a = 8 - b ------ eq (i)



A decrease in length, by .5 results in new length being: a - .5

An increase in width, by 4 results in new width being: b + 4

If new area is doubled, then new area becomes: 2ab 

Therefore, (a - .5)(b + 4) = 2ab

ab + 4a - .5b - 2 = 2ab

2ab - ab - 4a + .5b = - 2

ab - 4a + .5b = - 2 ------- eq (ii)



b(8 - b) - 4(8 - b) + .5b = - 2 ------ Substituting 8 - b for a on eq (ii)

8b+-+b%5E2+-+32+%2B+4b+%2B+.5b+=+-+2

-+b%5E2+%2B+8b+%2B+4b+%2B+.5b+-+32+=+-+2

-+b%5E2+%2B+12.5b+-+32+=+-+2

b%5E2+-+12.5b+%2B+32+-+2+=+0

b%5E2+-+12.5b+%2B+30+=+0

2b%5E2+-+25b+%2B+60+=+0 ------- Multiplying by 2 to clear decimal

Using the quadratic equation,

width, or highlight_green%28b+=+3.2396%29  OR   b+=+9.260398645 (ignore as a = 8 - b, which would result in a negative value for a)

a = 8 - 3.2396 ------- Substituting 3.2396 for b in eq (i)

Length, or highlight_green%28a+=+4.7604%29

You can do the check!! 



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Further help is available, online or in-person, for a fee, obviously. 



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