Find the dimensions of the rectangular box given the volume and relationships between its dimensions.
Write the equation representing the relationship.
The height of the box is 8ft, and the length L is 2ft longer than thrice the width x. The volume V is given by the equation:
V=L⋅W⋅H=(3x+2)⋅x⋅8=680
This simplifies to:
24x2+16x−680=0
Factor the polynomial.
The polynomial 24x2+16x−680 can be factored as:
8(x−5)(3x+17)=0
Solve for x.
Setting each factor to zero gives:
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
3x+17=0⇒x=−317(not valid since width cannot be negative)
Thus, the width is:
x=5ft
Calculate the length.
Using the width to find the length:
L=3x+2=3(5)+2=15+2=17ft
Thus, the length is:
L=17ft
The width of the box is 5ft and the length is 17ft.
The width of the box is 5ft.
The length of the box is 17ft.