A gift box has a value of 450 in.³. The width of the box is 4 inches less than the length. The height is twice the witdth. what is the width and inches of the gift box?
To find the dimensions of the gift box, we can use the given information about its volume and the relationships between the dimensions.
Let's start by assigning variables to represent the unknowns:
Length of the box = L inches
Width of the box = W inches
Height of the box = H inches
Given information:
Volume of the box = 450 in³
The volume of a rectangular box is calculated by multiplying its length, width, and height:
Volume = Length * Width * Height
We are also given the following relationships:
Width = Length - 4
Height = 2 * Width
Using these relationships, we can substitute the expressions for width and height into the volume formula:
450 = L * (L - 4) * (2 * (L - 4))
Simplifying the equation:
450 = L * (L - 4) * (2L - 8)
450 = L * (2L² - 16L + 32)
450 = 2L³ - 16L² + 32L
Rearranging the equation to isolate the cubic term:
2L³ - 16L² + 32L - 450 = 0
Now, we can solve this equation using a numerical method or calculator to find the value of L. Once we find L, we can substitute it back into the other equations to find the values of W and H.
After solving the equation, we find that L ≈ 9.917 inches.
Using the given relationships:
Width = L - 4 ≈ 9.917 - 4 ≈ 5.917 inches
Height = 2 * Width ≈ 2 * 5.917 ≈ 11.834 inches
Therefore, the approximate dimensions of the gift box are:
Length ≈ 9.917 inches
Width ≈ 5.917 inches
Height ≈ 11.834 inches
L = w+4
h = 2w
450 = w L h
450 = w * (w+4) * 2 w
450 = 2 w^2 (w+4) = 2 w^3 + 8 w^2
w^3 + 4 w = 225
w^2 (w+4) = 225
guess w around 200^.3 or 4.9
then w^2(w+4) = 213.8 not bad guess
try w = 5
25(9) = 225, we are there
check
w = 5
L = 9
h = 10
sure enough