Question

1. balloons salina is having a surprise party for her friend ernie. the table shows how many balloons she has been able to blow up by the end of each 10 - minute segment.

time (min)	10	20	30	40	50

a. construct a scatter plot of the data. then draw and assess a line that seems to best represent the data.

b. use the line of best fit to make a conjecture about the number of balloons she will have blown up at the end of 70 minutes.

Explanation:

Part a

Step1: Plot the data points

For each time (x - axis: 10, 20, 30, 40, 50) and corresponding balloons (y - axis: 3, 12, 15, 16, 21), plot the points \((10,3)\), \((20,12)\), \((30,15)\), \((40,16)\), \((50,21)\) on the scatter plot (with x - axis as time in minutes and y - axis as number of balloons).

Step2: Draw the line of best fit

Visually, a line that approximately balances the number of points above and below it. Looking at the data, a line with a positive slope (since as time increases, balloons increase) can be drawn. For example, a line passing close to the points, maybe with a slope calculated as \(\frac{21 - 3}{50 - 10}=\frac{18}{40}=0.45\) (approximate), and y - intercept around when \(x = 0\), maybe \(y=-1.5\) (so equation \(y = 0.45x-1.5\)). Assessing the line: it should pass through the general trend of the data, showing a positive linear relationship (weak to moderate, since the points are somewhat spread but have an upward trend).

Part b

Step1: Determine the line of best fit equation

Using the two - point formula or visual estimation. Let's use two points, say \((10,3)\) and \((50,21)\). The slope \(m=\frac{21 - 3}{50 - 10}=\frac{18}{40}=0.45\). Using point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(10,3)\), we get \(y - 3=0.45(x - 10)\), which simplifies to \(y=0.45x - 4.5 + 3=0.45x-1.5\).

Step2: Predict for \(x = 70\)

Substitute \(x = 70\) into the equation \(y=0.45x-1.5\). So \(y=0.45\times70-1.5=31.5 - 1.5 = 30\). (Alternatively, using a visual line of best fit, if we see the trend, at 60 minutes (not in data), it would be around \(0.45\times60 - 1.5=27 - 1.5 = 25.5\), and at 70, 30 is a reasonable estimate.)

Answer:

s:
a. Scatter plot with points \((10,3)\), \((20,12)\), \((30,15)\), \((40,16)\), \((50,21)\) and a line of best fit with positive slope (equation approx \(y = 0.45x-1.5\)) showing a positive linear trend.
b. \(\boxed{30}\) (approximate, depending on the line of best fit, could be slightly different, but around 30)