The dimensions, in inches, of a shipping box at We Ship 4 You can be expressed as width x, length x + 5, and height 3x - 1. The volume is about 7.6 ft. Find the dimensions of the box in inches. Round to the nearest inch.
We need to convert the volume from cubic feet to cubic inches, and then solve for x. Since 1 cubic foot is equal to 12^3 = 1728 cubic inches, the volume in cubic inches is 7.6 ft * 1728 in^3/ft^3 = 13104 in^3.
The volume of a rectangular prism is given by the product of its length, width, and height. Therefore, we have the equation:
(width)(length)(height) = 13104
The dimensions are given by width x, length x + 5, and height 3x - 1. Substituting these expressions into the equation, we have:
(x)(x + 5)(3x - 1) = 13104
Expanding and simplifying, we get:
3x^3 + 14x^2 - 3x - 13104 = 0
To solve this equation, we can use numerical methods such as Newton's method or the bisection method. However, this might be too complex for manual calculations.
Alternatively, we can use approximations to find an approximate value for x. Since the volume is relatively small, we can assume that x is also small. Let's try x = 10:
(width)(length)(height) = (10)(15)(29) = 4350
This is significantly smaller than the actual volume of 13104 in^3. Let's try a larger value for x: x = 20:
(width)(length)(height) = (20)(25)(59) = 29500
This is larger than the actual volume. Therefore, we know that the value of x must be between 10 and 20.
Let's try x = 15:
(width)(length)(height) = (15)(20)(44) = 13200
This is closer to the actual volume of 13104 in^3. Therefore, the dimensions of the box in inches are approximately:
Width: x = 15 inches
Length: x + 5 = 20 inches
Height: 3x - 1 = 44 inches
Rounded to the nearest inch, the dimensions of the box in inches are:
Width: 15 inches
Length: 20 inches
Height: 44 inches