Question

on the given axes, sketch a graph of the area of each piece as a function of the number of cuts.

Explanation:

To solve this, we assume we start with a shape (e.g., a square or rectangle) with an initial area. Let's assume we start with a 10x10 square (area = 100 square inches). Each cut (assuming we cut the shape into equal - sized pieces each time) will increase the number of pieces. If we make \(n\) cuts, and we assume that each cut divides the number of pieces by a certain rule. For example, if we are cutting a square into equal - sized rectangles or smaller squares, and we make \(x\) cuts (let's say we are making parallel cuts along one side), the number of pieces \(N=x + 1\). Then the area of each piece \(A=\frac{\text{Initial Area}}{x + 1}\)

Step 1: Determine the function

Let the initial area \(A_0 = 100\) (assuming a 10x10 square). If the number of cuts is \(x\), and the number of pieces is \(x + 1\) (for example, 0 cuts: 1 piece, area 100; 1 cut: 2 pieces, area 50; 2 cuts: 3 pieces, area \(\frac{100}{3}\approx33.33\); 3 cuts: 4 pieces, area 25; 4 cuts: 5 pieces, area 20; 5 cuts: 6 pieces, area \(\frac{100}{6}\approx16.67\); 6 cuts: 7 pieces, area \(\frac{100}{7}\approx14.29\))

The function for the area \(y\) (in square inches) as a function of the number of cuts \(x\) is \(y=\frac{100}{x + 1}\), where \(x = 0,1,2,\cdots\)

Step 2: Plot the points

• When \(x = 0\), \(y=\frac{100}{0 + 1}=100\)
• When \(x = 1\), \(y=\frac{100}{1+1} = 50\)
• When \(x=2\), \(y=\frac{100}{2 + 1}\approx33.33\)
• When \(x = 3\), \(y=\frac{100}{3+1}=25\)
• When \(x=4\), \(y=\frac{100}{4 + 1}=20\)
• When \(x = 5\), \(y=\frac{100}{5+1}\approx16.67\)
• When \(x=6\), \(y=\frac{100}{6 + 1}\approx14.29\)

To sketch the graph:

1. Mark the points \((0,100)\), \((1,50)\), \((2,\frac{100}{3})\), \((3,25)\), \((4,20)\), \((5,\frac{100}{6})\), \((6,\frac{100}{7})\) on the coordinate plane where the x - axis is the number of cuts and the y - axis is the area of each piece.
2. Connect the points with a smooth curve (since the relationship is a reciprocal relationship \(y=\frac{100}{x + 1}\), the graph will be a hyperbola - like curve in the first quadrant, with the x - axis as a horizontal asymptote as \(x

ightarrow\infty\) and the y - axis as a vertical asymptote as \(x
ightarrow - 1\), but since \(x\geq0\) (number of cuts can't be negative), we only consider \(x\geq0\))

The graph will show a decreasing curve as the number of cuts increases, since as we make more cuts, the area of each individual piece decreases.

Answer:

To solve this, we assume we start with a shape (e.g., a square or rectangle) with an initial area. Let's assume we start with a 10x10 square (area = 100 square inches). Each cut (assuming we cut the shape into equal - sized pieces each time) will increase the number of pieces. If we make \(n\) cuts, and we assume that each cut divides the number of pieces by a certain rule. For example, if we are cutting a square into equal - sized rectangles or smaller squares, and we make \(x\) cuts (let's say we are making parallel cuts along one side), the number of pieces \(N=x + 1\). Then the area of each piece \(A=\frac{\text{Initial Area}}{x + 1}\)

Step 1: Determine the function

Let the initial area \(A_0 = 100\) (assuming a 10x10 square). If the number of cuts is \(x\), and the number of pieces is \(x + 1\) (for example, 0 cuts: 1 piece, area 100; 1 cut: 2 pieces, area 50; 2 cuts: 3 pieces, area \(\frac{100}{3}\approx33.33\); 3 cuts: 4 pieces, area 25; 4 cuts: 5 pieces, area 20; 5 cuts: 6 pieces, area \(\frac{100}{6}\approx16.67\); 6 cuts: 7 pieces, area \(\frac{100}{7}\approx14.29\))

The function for the area \(y\) (in square inches) as a function of the number of cuts \(x\) is \(y=\frac{100}{x + 1}\), where \(x = 0,1,2,\cdots\)

Step 2: Plot the points

• When \(x = 0\), \(y=\frac{100}{0 + 1}=100\)
• When \(x = 1\), \(y=\frac{100}{1+1} = 50\)
• When \(x=2\), \(y=\frac{100}{2 + 1}\approx33.33\)
• When \(x = 3\), \(y=\frac{100}{3+1}=25\)
• When \(x=4\), \(y=\frac{100}{4 + 1}=20\)
• When \(x = 5\), \(y=\frac{100}{5+1}\approx16.67\)
• When \(x=6\), \(y=\frac{100}{6 + 1}\approx14.29\)

To sketch the graph:

1. Mark the points \((0,100)\), \((1,50)\), \((2,\frac{100}{3})\), \((3,25)\), \((4,20)\), \((5,\frac{100}{6})\), \((6,\frac{100}{7})\) on the coordinate plane where the x - axis is the number of cuts and the y - axis is the area of each piece.
2. Connect the points with a smooth curve (since the relationship is a reciprocal relationship \(y=\frac{100}{x + 1}\), the graph will be a hyperbola - like curve in the first quadrant, with the x - axis as a horizontal asymptote as \(x

ightarrow\infty\) and the y - axis as a vertical asymptote as \(x
ightarrow - 1\), but since \(x\geq0\) (number of cuts can't be negative), we only consider \(x\geq0\))

The graph will show a decreasing curve as the number of cuts increases, since as we make more cuts, the area of each individual piece decreases.