math question?

Question

A photographer has a picture that is 6 inches by 8 inches. he wishes to reduce the same amount on each side such that the resulting photo will have an area that is half the area of the original picture. by how much should each side be reduce?

Answer 1

(6-x)(8-x)=24
48-14x+x^2=24
x^2-14x+24=0
(x-12)(x-2)=0
only answer physically possible is x=2
the new photo will be 4*6=24in^2

Answer 2

If both sides are reduced by the same factor, the area changes as the square of that factor. The sides change as the square root of the area. The new dimension of each side is then √.5 (√2/2) times that side. = √2/2. The new dimensions will be 3√2 x 4√2. The new area is then 3√2 * 4√2 = 12*2 = 24, half the original area. This is if the sides are reduced by a factor and not by a fixed amount. It's another way to look at the problem.

NOTE: While I do not question that taking 2" from each side is the intended answer, it is not the way anyone in a practical situation would do it, because that changes the proportions of the original. The original aspect ratio (width to height) is 8/6 = 1.33; after taking two inches off, the aspect ratio is 6/4 = 1.5. The same picture won't fit into both frames without cutting something off.

Answer 3

Like everyone before me, reduce the sides by two inches each.

You want the resulting photo to have an area that's half the area of the original picture. The area of the original picture is 6x8=48. So, what's half of 48 is 24. So you know whatever length or width this resulting picture is, it must be a factor of 24. Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Well, it obviously can't be 1, 2, 12 or 24 because the original sides aren't so large to be reduced to these actual terms. So we're left with 3, 4, 6, and 8. Well, 3 and 8 are the original dimensions, and we want to REDUCE from that, so the only terms left are 4 and 6, which are both attainable by reduction in size and factors of 24.

Answer 4

6 x 8 = 48 sq inches

to reduce by half you want a picture with the dimensions:

(6-A) x (8-A) = 24

48 - 14 A + A^2 = 24

A^2 - 14A + 24 = 0

Using the quadratic formula:
A = 14 +- sq(14^2 - 96) / 2
A = 4/2 or 24 /2

Since A can not be greater than 6 or 8, A = 2
of course this will change the proportions of the picture to 4 x 6

Answer 5

it would be reduced to a 4 x 6 or it could be a 3 x 8

Answer 6

(6-x) X (8-x) = 24 sq in
(48 - 24) - 14x + x^2 = 0
x^2 - 14x + 24 = 0

(x - 12) (X - 2) = 0

x = 12 & 2

since using x=12 would cause unreal results, then x=2 is the only real value.

so,
6-2 = 4, 8-2=6
answer = 4" and 6"

Answer 7

reduce each side by 2 inches

Answer 8

(6-x)*(8-x)=6*8/2
This will give you (x-12)*(x-2)=0
from this you get x=2, x can't be 12 because it has to be smaller than 6.

Answer 9

6x8=48
48/2=24
4x6=24

each side decreases by 2