Question

1. a group of students conducted the bridge-thickness experiment with construction paper. the table below contains their results.

bridge-thickness experiment
number of layers | 1 | 2 | 3 | 4 | 5 | 6
breaking weight (pennies) | 12 | 20 | 29 | 42 | 52 | 61
a. make a graph of the (number of layers, breaking weight) data. describe the relationship between breaking weight and number of layers.
b. suppose it is possible to use half-layers of construction paper. what breaking weight would you predict for a bridge 3.5 layers thick? explain.
c. predict the breaking weight for a construction-paper bridge of 8 layers. explain how you made your prediction.

Explanation:

Part a

To make the graph, we use the number of layers as the x - axis (independent variable) and breaking weight (in pennies) as the y - axis (dependent variable). We plot the points \((1,12)\), \((2,20)\), \((3,29)\), \((4,42)\), \((5,52)\), \((6,61)\). When we look at the pattern of the points, as the number of layers (x - value) increases, the breaking weight (y - value) also increases. The relationship appears to be approximately linear (we can check the differences between consecutive y - values: \(20 - 12=8\), \(29 - 20 = 9\), \(42-29 = 13\), \(52 - 42=10\), \(61 - 52 = 9\) - the differences are relatively close, so a linear model is a reasonable approximation).

Step 1: Find the linear regression equation (or use the average rate of change)

First, we can find the slope between two points. Let's use the first two points \((1,12)\) and \((6,61)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{61 - 12}{6 - 1}=\frac{49}{5}=9.8\). We can also use the point - slope form \(y - y_1=m(x - x_1)\). Using the point \((1,12)\), the equation is \(y-12 = 9.8(x - 1)\), which simplifies to \(y=9.8x+2.2\).

Step 2: Predict for \(x = 3.5\)

Substitute \(x = 3.5\) into the equation \(y=9.8x + 2.2\). \(y=9.8\times3.5+2.2=34.3 + 2.2=36.5\). Alternatively, we can use the average rate of change between \(x = 3\) (breaking weight = 29) and \(x = 4\) (breaking weight = 42). The rate of change between \(x = 3\) and \(x = 4\) is \(42-29 = 13\) per layer. For a half - layer (from \(x = 3\) to \(x = 3.5\)), we can approximate the increase as \(\frac{13}{2}=6.5\). So the predicted breaking weight is \(29 + 6.5=35.5\). The linear regression gives a more accurate result, but both methods are reasonable. The linear regression equation gives us a prediction of approximately 36.5 pennies.

Step 1: Use the linear regression equation

We have the linear regression equation \(y=9.8x + 2.2\) from part b.

Step 2: Substitute \(x = 8\) into the equation

Substitute \(x = 8\) into \(y=9.8x+2.2\). \(y=9.8\times8+2.2=78.4+2.2 = 80.6\). We can also check the pattern of the data. The differences between consecutive breaking weights are relatively consistent (around 9 - 13). From \(x = 6\) (breaking weight = 61), the rate of change from \(x = 5\) to \(x = 6\) is \(61 - 52=9\). If we assume a similar rate of change, from \(x = 6\) to \(x = 7\), we add 9 (approximate) to get \(61 + 9=70\), and from \(x = 7\) to \(x = 8\), we add 9 (approximate) to get \(70+9 = 79\). The linear regression gives a more precise prediction of approximately 80.6 pennies.

Answer:

• Graph: A scatter plot with x - axis labeled "Number of Layers" (values 1 - 6) and y - axis labeled "Breaking Weight (pennies)" (values from 0 to 70 approximately). Plot the points \((1,12)\), \((2,20)\), \((3,29)\), \((4,42)\), \((5,52)\), \((6,61)\).
• Relationship: As the number of layers of construction paper increases, the breaking weight (in pennies) generally increases, and the relationship is approximately linear.

Part b