Marianne and faith are upgrading the storage shed in their backyard. the existing shed has the shape of a rectangular prism with dimensions 1 m by 2 m by 4 m. By what amount should they increase each dimension so that the new shed is 9 times the volume of the original shed? Provide a full algebraic solution.
volume grows as the scale cubed, so multiply each dimension by ∛9
To see this, suppose each dimension is multiplied by k. Then you want
(kx)(ky)(kz) = 9xyz
k^3 xyz = 9xyz
k^3 = 9
Good luck getting the right tape measure!
To solve this problem algebraically, let's first calculate the volume of the original shed. The volume of a rectangular prism is found by multiplying its length, width, and height together.
Given that the original shed has dimensions 1 m by 2 m by 4 m, its volume is:
Original volume = 1 m * 2 m * 4 m = 8 cubic meters
Let's call the amount by which each dimension is increased as 'x.' Therefore, the new dimensions of the shed will be:
Length = 1 + x meters
Width = 2 + x meters
Height = 4 + x meters
Using these new dimensions, the volume of the new shed will be:
New volume = (1 + x) meters * (2 + x) meters * (4 + x) meters
According to the problem statement, the new volume should be 9 times the original volume. So we can write the following equation to solve for x:
New volume = 9 * Original volume
(1 + x) meters * (2 + x) meters * (4 + x) meters = 9 * 8 cubic meters
Now we can simplify the equation and solve for x:
(1 + x)(2 + x)(4 + x) = 72
Expanding the left side of the equation:
(2 + 3x + x^2)(4 + x) = 72
Multiplying both sides by (4 + x):
(2 + 3x + x^2)(4 + x) = 72
8 + 2x + 12x + 3x^2 + 4x + x^2 = 72
Now, combine like terms:
12 + 18x + 4x^2 = 72
Rearranging the equation:
4x^2 + 18x + 12 = 72
Now, subtract 72 from both sides:
4x^2 + 18x - 60 = 0
Divide through by 2 to simplify the equation:
2x^2 + 9x - 30 = 0
This quadratic equation can be factored:
(2x - 5)(x + 6) = 0
Setting each factor to zero and solving for x:
2x - 5 = 0 -> 2x = 5 -> x = 5/2
x + 6 = 0 -> x = -6
We discard the negative value (-6) since dimensions cannot be negative.
Therefore, x = 5/2.
To find the amount by which each dimension should be increased, we substitute this value back into the length, width, and height expressions:
Length = 1 + x = 1 + (5/2) = 7/2 = 3.5 meters
Width = 2 + x = 2 + (5/2) = 9/2 = 4.5 meters
Height = 4 + x = 4 + (5/2) = 13/2 = 6.5 meters
So, they should increase each dimension by 3.5 meters, 4.5 meters, and 6.5 meters, respectively.