We appreciate your visit to 18 18 18 99 2i 6 6 8 An open rectangular box is to be cut out from a sheet of cardboard that has dimensions. 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 :
Final Answer:
The volume of the open rectangular box expressed as a polynomial is: V(x) = x(18 - 2x)(14 - 2x).
Explanation:
To find the volume of the open rectangular box, we need to consider the dimensions of the sheet of cardboard and the length of the corners being cut out. The dimensions of the cardboard are given as 18cm and 14cm. Let's denote the length of each cut corner as 'x'. When we cut out corners of length 'x' from each corner of the sheet and fold it up, the dimensions of the resulting box would be (18 - 2x) × (14 - 2x) × x.
The volume of a rectangular box is given by V = l × w × h, where l is the length, w is the width, and h is the height. In this case, the length of the box is (18 - 2x), the width is (14 - 2x), and the height is x. Multiplying these dimensions gives us the expression for the volume: V(x) = x(18 - 2x)(14 - 2x).
To sketch a graph of the polynomial, we plot the x-axis for the values of 'x' and the y-axis for the values of 'V(x)' (volume). The x-intercepts are where the volume is zero, which occurs when either x = 0 or when the terms inside the parentheses equal zero (18 - 2x = 0 or 14 - 2x = 0). Solving these equations gives us the critical points. The y-intercept occurs when x = 0. Plotting these points and connecting them will give us the graph of the polynomial.
The domain for 'x' is determined by the constraints of the problem. The length of each cut corner cannot exceed half the dimensions of the cardboard (x ≤ 7) to ensure the box can be formed. Also, x should be non-negative (x ≥ 0). Therefore, the domain for x is 0 ≤ x ≤ 7.
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