We appreciate your visit to 6 2 The Return of the Box Earlier we learned we can make a box from a piece of paper by cutting squares of side. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
The leading term of the polynomial is [tex]4x^{3}[/tex] and the degree of the polynomial is 3.
A sheet of paper measuring 8.5 inches by 14 inches is currently in our possession. The box's volume V, measured in cubic inches, depends on the side length x, where V (x) = (14-2x)(8.5-2x)(x).
So, we have :
V (x) = (14-2x)(8.5-2x)(x)
[tex]V(x) = (14-2x)(8.5-2x)(x).\\V (x) = (14-2x)(8.5x-2x^{2} )\\V(x) = 4x^{3} -45x^{2} + 119x[/tex]
Thus, the leading term of the polynomial is [tex]4x^{3}[/tex] and the degree of the polynomial is 3.
Now, we have :
[tex]V(x) = 4x^{3} -45x^{2} + 119x\\V(x) = x ( 4x^{2} -45x +119)\\V(x) = x (x-7)(4x-17)[/tex]
Now, For intercepts, V(x) = 0
[tex]V(x) = x (x-7)(4x-17) = 0\\x (x-7)(4x-17) = 0\\\\x = 0, 7, 4.25[/tex]
Now when x = 0, V(x) =0
when x = 7, f(x) = 135
when x = 4.25, f(x) = 252.89
So, the points are (7, 135) and ( 4.25, 252.89)
These points represent the value of x and their corresponding value of V.
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Final answer:
The polynomial is of degree 3 and the leading term is -4x^3. The horizontal intercepts are at x = 0, x = 7, and x = 4.25, representing the lengths of cuts that result in zero volume. The vertical intercept of the graph is at y = 0.
Explanation:
The function V(x) = (14-2x)(8.5-2x)(x) represents the volume of the box. This is a trinomial equation, therefore its degree is 3 and its leading term is -4x^3.
The horizontal intercept (or x-intercept) is the value of x for which V(x) = 0. From the equation, we can see that V(x) = 0 when x = 0, x = 7, and x = 4.25. This implies that there are horizontal intercepts at x = 0, x = 7, and x = 4.25.
As for the vertical intercept (or y-intercept), it is the value of V(x) when x = 0. Substituting x = 0 in the equation, V(x) = (14)(8.5)(0) = 0. So, the vertical intercept is at y = 0. In the context of making a box, these x-intercepts make sense as they represent the possible lengths of cuts for which the box will have a volume of zero.
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