Given the volume of a rectangular prism is [tex]\( V = l \cdot w \cdot h \)[/tex], where [tex]\( l \)[/tex] is the length, [tex]\( w \)[/tex] is the width, and [tex]\( h \)[/tex] is the height, let's set up the equation based on the given conditions:
1. The height [tex]\( h \)[/tex] of the tray is 3 centimeters.
2. The volume [tex]\( V \)[/tex] of the tray is 252 cubic centimeters.
3. The length [tex]\( l \)[/tex] of the tray is 5 centimeters longer than its width [tex]\( w \)[/tex].
Let [tex]\( x \)[/tex] represent the width of the tray. Then, the length [tex]\( l \)[/tex] can be expressed as [tex]\( x + 5 \)[/tex].
Using these values in the volume formula, we get:
[tex]\[ 252 = (x + 5) \cdot x \cdot 3 \][/tex]
Simplify the equation:
[tex]\[ 252 = 3(x^2 + 5x) \][/tex]
Divide both sides by 3 to isolate the quadratic expression:
[tex]\[ 84 = x^2 + 5x \][/tex]
Rewriting this in standard quadratic equation form, we have:
[tex]\[ x^2 + 5x - 84 = 0 \][/tex]
So, the equation that models the volume of the tray in terms of its width [tex]\( x \)[/tex] is:
[tex]\[ x^2 + 5x - 84 = 0 \][/tex]
For verification, we know that the coefficients of this equation are:
- [tex]\( A = 1 \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( B = 5 \)[/tex] (the coefficient of [tex]\( x \)[/tex])
- [tex]\( C = -84 \)[/tex] (the constant term)
Now, determining whether a width [tex]\( x = 7.5 \)[/tex] is a possible solution involves checking if it satisfies this quadratic equation. Substituting [tex]\( x = 7.5 \)[/tex] into the equation:
[tex]\[ (7.5)^2 + 5(7.5) - 84 \][/tex]
Calculating each term:
[tex]\[ 7.5^2 = 56.25 \][/tex]
[tex]\[ 5 \cdot 7.5 = 37.5 \][/tex]
[tex]\[ 56.25 + 37.5 - 84 = 93.75 - 84 = 9.75 \][/tex]
Since substituting [tex]\( x = 7.5 \)[/tex] does not result in zero, [tex]\( 7.5 \)[/tex] is not a solution of the equation. Thus, it is not possible for the width of the tray to be 7.5 centimeters.
So the final answers are:
Complete the equation that models the volume of the tray in terms of its width, [tex]\( x \)[/tex], in centimeters:
[tex]\[ x^2 + 5x = 84 \][/tex]
Is it possible for the width of the tray to be 7.5 centimeters?
[tex]\[ \text{No} \][/tex]
