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=W h \\
P=2(1+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)
Express the perimeter of the rectangular base as 2 ( l + w ) = 28 , and solve for l in terms of w , obtaining l = 14 − w .
Substitute l = 14 − w and h = 5 into the volume equation V = lw h , resulting in V = 5 ( 14 − w ) w .
Simplify the volume equation to V = 70 w − 5 w 2 , which can be written as V ( w ) = − 5 w 2 + 70 w .
Compare the derived equation with the given options and identify the matching equation: y = 5 ( 14 − x ) ( x ) .
Explanation
Problem Analysis Let's analyze the problem. We are given a box with a rectangular base and a height of 5 cm. The perimeter of the rectangular base is 28 cm. We need to find a quadratic equation that models the volume of the box.
Express length in terms of width Let l be the length and w be the width of the rectangular base. The perimeter P is given by: P = 2 ( l + w ) = 28 So, l + w = 14 , which means l = 14 − w .
Calculate the volume The volume V of the box is given by: V = lw h Since the height h = 5 , we have: V = ( 14 − w ) w ( 5 ) V = 5 ( 14 − w ) w V = 5 ( 14 w − w 2 ) V = 70 w − 5 w 2
Find the matching equation So the volume V as a function of w is: V ( w ) = − 5 w 2 + 70 w Now let's compare this with the given options, replacing w with x and V ( w ) with y :
y = 5 ( 28 − x ) ( x ) expands to y = 5 ( 28 x − x 2 ) = 140 x − 5 x 2 y = 5 ( 28 − 2 x ) ( x ) expands to y = 5 ( 28 x − 2 x 2 ) = 140 x − 10 x 2 y = 5 ( 14 − x ) ( x ) expands to y = 5 ( 14 x − x 2 ) = 70 x − 5 x 2 y = 5 ( 14 − 2 x ) ( x ) expands to y = 5 ( 14 x − 2 x 2 ) = 70 x − 10 x 2 The equation that matches our derived equation is y = 5 ( 14 − x ) ( x ) .
Final Answer Therefore, the quadratic equation that best models the volume of the box is: y = 5 ( 14 − x ) ( x )
Examples
Understanding how to model the volume of a box with a given perimeter is useful in various real-world scenarios. For example, if you are designing a rectangular garden bed with a limited amount of fencing (perimeter), you can use this model to determine the dimensions that maximize the volume (and thus the amount of soil) within the garden bed. This helps optimize the use of available resources and space, whether you're gardening, packaging products, or designing storage containers.
The quadratic equation that models the volume of the box is given by C: y = 5 ( 14 − x ) ( x ) . This is derived by using the perimeter to express the length in terms of width and substituting it into the volume formula. The final equation accurately reflects the relationship between the width and volume of the box.
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