Garfield has designed a rectangular storage unit to hold large factory equipment. His scale
model has dimensions 1 m by 2 m by 4 m. In order to maintain the scale he needs to
increase each dimension by the same amount. By what amount should he increase each
dimension to produce an actual storage unit that is no more than 9 times the volume of his
scale model?
To solve this problem, we need to find the amount by which Garfield should increase each dimension of the scale model to produce an actual storage unit that is no more than 9 times the volume of the scale model.
Let's start by calculating the volume of the scale model. The volume of a rectangular prism is found by multiplying its length, width, and height. In this case, the dimensions of the scale model are 1 m, 2 m, and 4 m:
Volume of scale model = length * width * height
= 1 m * 2 m * 4 m
= 8 cubic meters
Now, we need to determine the maximum volume for the actual storage unit. According to the problem, the actual storage unit should be no more than 9 times the volume of the scale model. Therefore, the maximum volume will be:
Maximum volume = 9 * volume of scale model
= 9 * 8 cubic meters
= 72 cubic meters
Now, we can find the dimensions of the actual storage unit. Since all three dimensions are increased by the same amount, let's assume the increase in each dimension is 'x' meters.
The dimensions of the actual storage unit will be:
Length = 1 m + x
Width = 2 m + x
Height = 4 m + x
And we need to find 'x'. The volume of the actual storage unit will be:
Volume of actual storage unit = (1 m + x) * (2 m + x) * (4 m + x)
According to the problem, the volume of the actual storage unit should not exceed 72 cubic meters. So we can set up the following equation:
(1 m + x) * (2 m + x) * (4 m + x) ≤ 72
Now, we can solve this equation to find the maximum value of 'x'. Since this is a quadratic equation, we need to set it equal to zero first:
(1 m + x) * (2 m + x) * (4 m + x) - 72 = 0
To find the solution for 'x', we can solve this equation using factoring, the quadratic formula, or numerical methods like approximation. However, since finding the exact solution through factoring or the quadratic formula might be complicated, let's use numerical approximation.
Using an online graphing calculator or software, you can plot the function f(x) = (1+x)(2+x)(4+x) - 72 and find the value of 'x' that satisfies the equation. This will give you the amount by which Garfield should increase each dimension to produce an actual storage unit that is no more than 9 times the volume of his scale model.
the volume increases by the cube of the linear scale. So, he needs a scale factor of no more than
∛9 = 2.08