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The volume of a box, a rectangular prism, is represented by the function [tex]f(x)=x^3-x^2-17x-15[/tex]. The length of the box is [tex](x-5)[/tex], and the width is [tex](x+1)[/tex]. Find the expression representing the height of the box.

A. [tex]x-1[/tex]
B. [tex]x-3[/tex]
C. [tex]x+5[/tex]
D. [tex]x+3[/tex]

Sagot :

GinnyAnswer
To find the height of the box given its volume, length, and width, we will use the relationship:

[tex]\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \][/tex]

In this case, the volume [tex]\( f(x) \)[/tex] is [tex]\( x^3 - x^2 - 17x - 15 \)[/tex]. The length is [tex]\( (x-5) \)[/tex] and the width is [tex]\( (x+1) \)[/tex]. Let's denote the height by [tex]\( h(x) \)[/tex].

1. First, express the volume in terms of length, width, and height:

[tex]\[ f(x) = (x - 5)(x + 1)h(x) \][/tex]

2. Substitute the given expression for [tex]\( f(x) \)[/tex]:

[tex]\[ x^3 - x^2 - 17x - 15 = (x - 5)(x + 1)h(x) \][/tex]

3. To find [tex]\( h(x) \)[/tex], we need to solve for [tex]\( h(x) \)[/tex]. Divide both sides of the equation by the product of the length and width:

[tex]\[ h(x) = \frac{x^3 - x^2 - 17x - 15}{(x - 5)(x + 1)} \][/tex]

4. Next, we will perform polynomial long division to divide [tex]\( x^3 - x^2 - 17x - 15 \)[/tex] by [tex]\( (x - 5)(x + 1) \)[/tex].

Let’s find the product [tex]\( (x - 5)(x + 1) \)[/tex]:

[tex]\[ (x - 5)(x + 1) = x^2 + x - 5x - 5 = x^2 - 4x - 5 \][/tex]

Now, we need to divide [tex]\( x^3 - x^2 - 17x - 15 \)[/tex] by [tex]\( x^2 - 4x - 5 \)[/tex].

Perform polynomial long division:

1. [tex]\( x^3 \div x^2 = x \)[/tex]

2. Multiply [tex]\( x \)[/tex] by [tex]\( x^2 - 4x - 5 \)[/tex]:

[tex]\[ x \cdot (x^2 - 4x - 5) = x^3 - 4x^2 - 5x \][/tex]

3. Subtract this product from the original polynomial:

[tex]\[ (x^3 - x^2 - 17x - 15) - (x^3 - 4x^2 - 5x) = 3x^2 - 12x - 15 \][/tex]

4. Now divide the new polynomial [tex]\( 3x^2 - 12x - 15 \)[/tex] by [tex]\( x^2 - 4x - 5 \)[/tex]:

[tex]\[ 3x^2 \div x^2 = 3 \][/tex]

5. Multiply [tex]\( 3 \)[/tex] by [tex]\( x^2 - 4x - 5 \)[/tex]:

[tex]\[ 3 \cdot (x^2 - 4x - 5) = 3x^2 - 12x - 15 \][/tex]

6. Subtract this result from [tex]\( 3x^2 - 12x - 15 \)[/tex]:

[tex]\[ (3x^2 - 12x - 15) - (3x^2 - 12x - 15) = 0 \][/tex]

The division results in no remainder, so:

[tex]\[ h(x) = x + 3 \][/tex]

Thus, the height of the box is represented by the expression:

[tex]\[ x + 3 \][/tex]
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