A rectangular counter is covered with 600 square tiles.The counter could have been covered with 400 tiles that are 1 cm longer on each side (those are square too). Find the length of the side of the smaller tile.
a = the area (in equation)
Now, I know that the starting equation is 600 x a^2 = 400(a+1)^2.
But someone please tell me what each of thes numbers mean? What's the logic in this equation? Why do you need the areas?
Please, and thank you!
a = the area (in equation)
Now, I know that the starting equation is 600 x a^2 = 400(a+1)^2.
But someone please tell me what each of thes numbers mean? What's the logic in this equation? Why do you need the areas?
Please, and thank you!
-
Technically, the starting equation should be, if L = the length,
600 * L^2 = 400(L+1)^2
because L^2 and (L+1)^2 are the areas of each size tile. You shouldn't need the areas. Expand the right side and solve for L:
600 * L^2 = 400(L+1)^2
600 * L^2 = 400(L^2+2L+1)
600L^2 = 400L^2 + 800L + 400
Combine the quadratic terms and factor out anything that goes easily:
0 = -200L^2 + 800L + 400
0 = -200(L^2 - 4L - 2)
Divide both sides by -200:
0 = L^2 - 4L - 2
L = {-(-4) plus or minus [(-4)^2 - 4*1*-2]^0.5}/2*1
L = {4 plus or minus [16 + 8]^0.5}/2
L = {4 plus or minus [24]^0.5}/2
L = {4 plus or minus 2*[6]^0.5}/2
L = 2 plus or minus 6^0.5
The square root of 6 is just under 2.45, so the only answer that makes sense is 2 + 6^0.5, or about 4.45 cm; that's the length of one side of each of the shorter tiles.
600 * L^2 = 400(L+1)^2
because L^2 and (L+1)^2 are the areas of each size tile. You shouldn't need the areas. Expand the right side and solve for L:
600 * L^2 = 400(L+1)^2
600 * L^2 = 400(L^2+2L+1)
600L^2 = 400L^2 + 800L + 400
Combine the quadratic terms and factor out anything that goes easily:
0 = -200L^2 + 800L + 400
0 = -200(L^2 - 4L - 2)
Divide both sides by -200:
0 = L^2 - 4L - 2
L = {-(-4) plus or minus [(-4)^2 - 4*1*-2]^0.5}/2*1
L = {4 plus or minus [16 + 8]^0.5}/2
L = {4 plus or minus [24]^0.5}/2
L = {4 plus or minus 2*[6]^0.5}/2
L = 2 plus or minus 6^0.5
The square root of 6 is just under 2.45, so the only answer that makes sense is 2 + 6^0.5, or about 4.45 cm; that's the length of one side of each of the shorter tiles.