# Find the sides of a rectangular prism if the volume = 30 cm^3 , the height = 3 cm and the perimeter of the base = 14 cm.

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The formula for the volume of a rectangular prism is V = lwh

The formula for the perimeter of a rectangle is P = 2w + 2l

So,

30 = lw*3 and 14 = 2w + 2l

10/w = l l = 7 - w

--> 10/w = 7 - w

10 = 7w - w^2

0 = w^2 - 7w + 10

0 = (w-5) (w-2)

w = 5 or w = 2 --> l = 2 or 5

So, let w = 5 and l = 2

Let the sides be:

x, y and height h

We know that the volume:

V = x*y*h

30 = x*y*3

==> xy = 10 ..........(1)

We also know that the perimeter of the base is:

P = 2*x + 2*y = 14

==> 2x + 2y = 14 ...........(2)

Now let us substitute:

2 (10/y) + 2y = 14

20/y + 2y = 14

Multiply by y:

==> 20 + 2y^2 = 14y

==> 2y^2 - 14y + 20 = 0

Divide by 2:

==> y^2 -7y + 10 = 0

==> (y-5)(y-2) = 0

==> y1= 5 ==> x1= 2

==> y2= 2 ==> x2= 5

Then the measures of the side of the base are 2 and 5.

Volume of a rectangular prism is given by the formula:

Volume = (Area of the rectangular cross section)*Height

substituting given values of height and volume in the above formula:

30 = (Area of the rectangular cross section)*3

Therefore:

Area of the rectangular cross section = 30/3 = 10 cm^2

Let lengths of two side of the triangle be x and y

Then:

Area of rectangle = x*y = 10

Therefore:

y = 10/x

And:

Perimeter of the rectangle = 2*(x + y) = 14

==> x + y = 7

Substituting values of y as 10/x in above equation:

x + 10/x = 7

Multiplying both sides by x, and shifting all the terms of equation on left hand side:

x^2 - 7x + 10 = 0

==> x^2 - 2x - 5x + 10 = 0

==> x(x - 2) - 5(x - 2) = 0

==> (x - 2)(x - 5) = 0

Therefore:

x = 2 or 5

Therefore sides of the rectangular prism are 2 cm and 5 cm.