Bob has designed a rectangular storage unit to hold large factory equipment. His scale model has dimensions 1 m by 2 m by 4 m. By what amount should he increase each dimension to produce an actual storage unit that is 9 times the volume of his scale model?
volumes of similar solids are proportional to the cube of their sides
so volume of original = 8m^3
volume of new is 72 m^3
so let's take the smallest dimension of 1 m , since it is a unit dimension.
8/72 = 1^3/x^3
8x^3 = 72
x^3 = 9^(1/3)
x = 9^(1/3) = appr 2.08
So, he should increase each dimenstion by a factor of 2.08
or
the new dimensions should be 2.08 by 4.16 by 8.32 m
check:
(2.08(4.16)(8.32) = 71.99 , not bad using 2 decimal accuracy
To determine the increase in each dimension, we need to calculate the scale factor by comparing the volumes of the scale model to the actual storage unit.
The scale model has a volume of 1 m × 2 m × 4 m = 8 cubic meters.
To create a storage unit that is 9 times the volume of the scale model, we multiply the scale model's volume by 9: 8 cubic meters × 9 = 72 cubic meters.
Let x be the increase factor for each dimension.
The volume of the actual storage unit can be represented as follows:
(1 m + x) × (2 m + x) × (4 m + x) = 72 cubic meters.
Expanding this equation, we have:
(1 × 2 × 4) + (1 × 2 × x) + (1 × 4 × x) + (2 × 4 × x) + (x^3) = 72.
8 + 2x + 4x + 8x + x^3 = 72.
Combining like terms, we have:
x^3 + 14x + 8 = 72.
Rearranging the terms, we get:
x^3 + 14x - 64 = 0.
We solve this cubic equation to find the value of x.
Using numerical methods or calculators, we find that x ≈ 2.7815.
Therefore, Bob should increase each dimension by approximately 2.7815 meters to produce a storage unit that is 9 times the volume of his scale model.
To find out by what amount Bob should increase each dimension, we need to first calculate the volume of his scale model storage unit and then determine the dimensions of the actual storage unit.
1. Calculate the volume of the scale model:
Volume = Length x Width x Height
Volume = 1 m x 2 m x 4 m
Volume = 8 cubic meters
2. Determine the desired volume of the actual storage unit:
The actual storage unit should have a volume that is 9 times the volume of the scale model.
Desired Volume = 9 x 8 cubic meters
Desired Volume = 72 cubic meters
3. Calculate the dimensions of the actual storage unit:
Since the volume of a rectangular cuboid is determined by multiplying its length, width, and height, and the volumes of similar shapes are directly proportional to the cubes of their corresponding dimensions, we can use this principle to find the increase factor for each dimension.
Let's assume the increase factor for each dimension is "x".
So, the dimensions of the actual storage unit will be:
Length = 1 m x "x" meters
Width = 2 m x "x" meters
Height = 4 m x "x" meters
Now, we can calculate the volume of the actual storage unit using these dimensions:
Volume = Length x Width x Height
Volume = (1 m x "x" meters) x (2 m x "x" meters) x (4 m x "x" meters)
Volume = 8 m^3 x "x"^3
In order to make the volume of the actual storage unit equal to 72 cubic meters, we can set up the following equation:
8 m^3 x "x"^3 = 72 m^3
4. Solve the equation to find the value of "x":
Divide both sides of the equation by 8 m^3:
"x"^3 = 72 m^3 / 8 m^3
"x"^3 = 9
Take the cube root of both sides of the equation to get the value of "x":
"x" = ∛9
"x" ≈ 2.08 (rounded to two decimal places)
5. Calculate the increase for each dimension:
Length increase = 1 m x (2.08 - 1) ≈ 1.08 m
Width increase = 2 m x (2.08 - 1) ≈ 2.16 m
Height increase = 4 m x (2.08 - 1) ≈ 4.32 m
Therefore, Bob should increase each dimension by approximately 1.08 meters in length, 2.16 meters in width, and 4.32 meters in height to produce an actual storage unit that is 9 times the volume of his scale model.