A box is to be constructed with a rectangular base and a height of 5 cm. If the rectangular base must have a perimeter of 28 cm, which quadratic equation best models the volume of the box?
[tex]
\begin{array}{l}
V=h w h \\
p=2(l+w)
\end{array}
[/tex]
A. [tex]y=5(28-x)(x)[/tex]
B. [tex]y=5(28-2 x)(x)[/tex]
C. [tex]y=5(14-x)(x)[/tex]
D. [tex]y=5(14-2 x)(x)[/tex]
Answer (2)
The quadratic equation that models the volume of the box, with a height of 5 cm and a perimeter of 28 cm for the rectangular base, is y = 5 ( 14 − x ) ( x ) . This corresponds to option C. It accurately captures the relationship between the width and length of the base.
;
Express the length l of the rectangular base in terms of the width w using the perimeter formula: l = 14 − w .
Substitute the expression for l into the volume formula V = lw h to get V = 5 ( 14 − w ) w .
Expand the expression to obtain the quadratic equation: V = 70 w − 5 w 2 .
Identify the quadratic equation that models the volume of the box: y = 5 ( 14 − x ) ( x ) .
Explanation
Define variables and knowns Let l and w be the length and width of the rectangular base, respectively. The height h of the box is given as 5 cm. The perimeter P of the rectangular base is 28 cm. The volume V of the box is given by V = lw h . We need to find a quadratic equation that models the volume of the box in terms of one variable.
Express length in terms of width The perimeter of the rectangular base is given by P = 2 ( l + w ) = 28 . We can solve for l in terms of w :
2 ( l + w ) = 28 l + w = 14 l = 14 − w
Substitute into volume equation Substitute l = 14 − w into the volume equation V = lw h :
V = ( 14 − w ) w ( 5 ) V = 5 ( 14 w − w 2 ) V = 70 w − 5 w 2
Final equation The quadratic equation that models the volume of the box is V = − 5 w 2 + 70 w . We can rewrite this as V = 5 ( 14 w − w 2 ) or V = 5 ( 14 − w ) w . Comparing this with the given options, we see that the equation y = 5 ( 14 − x ) ( x ) best models the volume of the box, where y represents the volume V and x represents the width w .
Examples
Understanding how to model the volume of a box with a quadratic equation can be useful in various real-world scenarios. For example, if you are designing a rectangular container with a fixed perimeter for the base and want to maximize the volume, you can use this equation to find the optimal dimensions. This is applicable in packaging design, where you want to use the least amount of material while maximizing the space inside the package. Similarly, in construction, when building a rectangular storage unit with a limited perimeter, you can use this model to determine the dimensions that provide the largest storage capacity.
