A rectangular package to be sent by a delivery service can have a maximum combined length (y) and girth (perimeter of its cross section) of 300 inches. Assume that the width and height are the same (x). Find the equation for the volume of the box in terms of x alone as an expanded polynomial.
well, you know that the cross-section is x^2 and the length is y, so
v = x^2y
but, you also know that
4x+y = 300
now you can eliminate y.
To find the equation for the volume of the box in terms of x alone, we first need to write expressions for the length, girth, and height in terms of x.
The length of the box (y) can be expressed as:
y = 2x + 2x + x
= 4x + x
= 5x
The girth of the box (perimeter of its cross section) is given by:
perimeter = 2(length + width)
= 2(5x + x)
= 2(6x)
= 12x
Now we can set up an equation using the given condition that the maximum combined length and girth is 300 inches:
5x + 12x = 300
Simplifying this equation, we get:
17x = 300
Now we can solve for x by dividing both sides of the equation by 17:
x = 300/17
Once we have the value of x, we can substitute it back into the expressions for the length, width, and height to find the volume.
Length (y) = 5x = 5 * (300/17)
Width = x = 300/17
Height = x = 300/17
Now we can calculate the volume by multiplying the length, width, and height:
Volume = Length * Width * Height
= (5 * (300/17)) * (300/17) * (300/17)
Simplifying further, we get the equation for the volume of the box in terms of x alone as an expanded polynomial:
Volume = (450000/289) * x^3