Question

the art club is raising money for a field trip to an art museum. the band is raising money for new instruments. the results of both fundraisers are shown in the table. art club fundraising (a graph) band fundraising the line of best fit for the band fundraising can be represented by the equation y = 35x + 200. estimate the line of best fit for the art club. then use the lines to compare the fundraising efforts. the dropdown had more money before the fundraising started. the dropdown raised more money each month. the dropdown will have more money after 7 months.

Explanation:

Step1: Find the y - intercept (initial amount)

The equation of the line of best fit for the band is \(y = 35x+200\). The y - intercept \(b = 200\). For the art club, when \(x = 0\) (before fundraising started), from the scatter - plot, the y - value (total amount) is approximately \(y=50\). So, \(200>50\), the band had more money before fundraising started.

Step2: Find the slope (amount raised per month)

The slope of the band's line \(m = 35\). For the art club, using two points \((0,50)\) and \((7,300)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Here, \(x_1 = 0,y_1 = 50,x_2=7,y_2 = 300\). Then \(m=\frac{300 - 50}{7-0}=\frac{250}{7}\approx35.71\). Since \(35.71>35\), the art club raised more money each month.

Step3: Calculate the amount after 7 months

For the band: Substitute \(x = 7\) into \(y = 35x+200\). \(y=35\times7 + 200=245+200=445\). For the art club: Using the slope - intercept form \(y=mx + b\) with \(m=\frac{250}{7}\approx35.71\) and \(b = 50\). \(y=\frac{250}{7}\times7+50=250 + 50=300\). Wait, no, using the two - point form correctly. The art club line (estimated): assume the line passes through \((0,50)\) and \((7,300)\) (from the scatter - plot). The equation is \(y-50=\frac{300 - 50}{7}(x - 0)\), \(y=\frac{250}{7}x+50\). When \(x = 7\), \(y=250 + 50=300\). But if we use a better - estimated line for the art club (passing through \((0,50)\) and \((6,250)\)). The slope \(m=\frac{250 - 50}{6}=\frac{200}{6}=\frac{100}{3}\approx33.33\). Equation \(y=\frac{100}{3}x+50\). When \(x = 7\), \(y=\frac{100}{3}\times7+50=\frac{700}{3}+50=\frac{700 + 150}{3}=\frac{850}{3}\approx283.33\). Wait, using the original two points \((0,50)\) and \((7,300)\) (from the scatter - plot visual estimate). The amount for the art club at \(x = 7\) is \(y = 300\), for the band \(y=35\times7+200=445\). But if we use a more accurate estimate for the art club line (assuming a line that is a better fit to the scatter - plot, say passing through \((0,50)\) and \((6,250)\)). Slope \(m=\frac{250 - 50}{6}=\frac{200}{6}=\frac{100}{3}\approx33.33\), \(y=\frac{100}{3}x+50\). When \(x = 7\), \(y=\frac{100}{3}\times7+50=\frac{700+150}{3}=\frac{850}{3}\approx283\). But if we use the visual estimate of the art club scatter - plot (the last point at \(x = 7\) is around \(y = 300\)). Band: \(y=35\times7+200=445\). Wait, no, re - check. The line of best fit for the art club (estimated from the scatter - plot): assume it passes through \((0,50)\) and \((7,300)\). The equation \(y-50=\frac{300 - 50}{7}(x - 0)\), \(y=\frac{250}{7}x + 50\). When \(x = 7\), \(y=250+50=300\). Band: \(y=35\times7+200=245 + 200=445\). But if we take a more conservative art club line (say passing through \((0,50)\) and \((6,250)\)). Slope \(m=\frac{250 - 50}{6}=\frac{200}{6}=\frac{100}{3}\approx33.33\), \(y=\frac{100}{3}x+50\). When \(x = 7\), \(y=\frac{700}{3}+50=\frac{700 + 150}{3}=\frac{850}{3}\approx283\). But looking at the art club scatter - plot, the points seem to follow a line that when \(x = 7\), \(y\approx300\). Band: \(y=35x+200\), when \(x = 7\), \(y = 35\times7+200=445\). But wait, no! Wait, the first two blanks: band (first blank), art club (second blank). For the third blank: band. Wait, no.
Wait, re - do:
- **Initial amount (\(x = 0\))**:
- Band: \(y=35(0)+200 = 200\)
- Art club (from scatter - plot \(x = 0,y = 50\)). So band had more before fundraising.
- **Slope (monthly raise)**:
- Band: \(m = 35\)
- Art club: Using two points \((0,50)\) and \((7,300)\), \(m=\frac{300 - 50}{7}=\frac{250}{7}\approx35.71\). Art club raised more each month.
- **Amount after 7 months**:
- Band: \(y=35\times7+200=245 + 200=445\)
- Art club: \(y=\frac{250}{7}\times7+50=250+50 = 300\) (using the two - point line). So band has more after 7 months.

Answer:

The band had more money before the fundraising started. The art club raised more money each month. The band will have more money after 7 months.