SOLUTION: The height and the base of a rectangle are positive integers. When the height is doubled and the base is tripled, the area of the new rectangle is 120 cm2. What is the smallest po

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Question 317720: The height and the base of a rectangle are positive integers. When the height is
doubled and the base is tripled, the area of the new rectangle is 120 cm2. What is the smallest possible perimeter of the smaller rectangle?

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let L and W be the length (base) and width (height) of the rectangle.
A1=L%2AW
.
.
.
A2=%283%2AL%29%2A%282W%29=6%2ALW=120
1.LW=20
The perimeter is
P=2%2A%28L%2BW%29=2L%2B2W
Use eq. 1, to get L as a function of W.
L=20%2FW
Suubstitute into the perimeter,
P=2%2820%2FW%29%2B2W
P=40%2FW%2B2W
Now P is a function of W only.
Take the derivative of P with respect to W and set it equal to zero to find the minimum.
dP%2FdW=-40%2FW%5E2%2B2=0
40%2FW%5E2=2
W%5E2=20
W=2%2Asqrt%285%29
Then from eq. 1,
L=20%2FW=20%2F%282%2Asqrt%285%29%29
L=10%2Fsqrt%285%29
L=2%2Asqrt%285%29
P=2%2AL%2B2%2AW=4%2A2%2Asqrt%285%29
highlight%28Pmin=8%2Asqrt%285%29%29