Question

this graph shows how the number of spools of thread that paula has left is related to the number of comforters that she makes. spools of thread remaining (graph with x-axis comforters made, y-axis spools of thread remaining, line from (0,5) to (8,0)) how many comforters can paula make, at most, and still have 2 spools of thread left? 6 comforters, 8 comforters, 5 comforters, 3 comforters

Explanation:

Step1: Analyze the graph

The graph is a line with x - axis (comforters made) and y - axis (spools of thread remaining). The y - intercept is 5 (when 0 comforters made, 5 spools left), and it goes to (8, 0) (8 comforters made, 0 spools left). The slope \(m=\frac{0 - 5}{8 - 0}=-\frac{5}{8}\). The equation of the line is \(y=-\frac{5}{8}x + 5\), where \(y\) is spools remaining and \(x\) is comforters made.

Step2: Find x when y = 2

Substitute \(y = 2\) into the equation:
\(2=-\frac{5}{8}x + 5\)
Subtract 5 from both sides: \(2-5=-\frac{5}{8}x\)
\(- 3=-\frac{5}{8}x\)
Multiply both sides by \(-\frac{8}{5}\): \(x=\frac{24}{5}=4.8\approx5\)? Wait, maybe better to look at the graph. When y = 2 (spools left), find the x - value. Looking at the grid, when spools remaining is 2, the x (comforters) is 6? Wait no, let's re - check. Wait the line goes from (0,5) to (8,0). Let's find the x when y = 2. The change in y from 5 to 2 is - 3. The total change in y is - 5 (from 5 to 0). The total change in x is 8. So the fraction of the way from y = 5 to y = 0 is \(\frac{3}{5}\). So x is \(8\times(1 - \frac{3}{5})=8\times\frac{2}{5}=3.2\)? No, maybe better to use the options. Let's check the options:
- For 6 comforters: Let's see the y - value. From the graph, when x = 6, what's y? The line from (0,5) to (8,0). The slope is \(-\frac{5}{8}\), so for x = 6, \(y=5-\frac{5}{8}\times6=5-\frac{30}{8}=5 - 3.75 = 1.25 eq2\)
- For 3 comforters: \(y=5-\frac{5}{8}\times3=5 - 1.875 = 3.125 eq2\)
- For 5 comforters: \(y=5-\frac{5}{8}\times5=5 - 3.125 = 1.875\approx2\)? Wait, maybe the graph is a bit approximate. Wait the options: 6,8,5,3. Wait when y = 2, looking at the graph, the x - value (comforters) is 6? No, wait the y - axis is spools remaining. Let's count the grid. Each square is 1 unit. When spools remaining is 2 (y = 2), the x (comforters) is 6? Wait no, let's look at the line. From (0,5) to (8,0). The points: (0,5), (1, 5 - 5/8=4.375), (2, 3.75), (3, 3.125), (4, 2.5), (5, 1.875), (6, 1.25), (7, 0.625), (8,0). Wait, maybe the question has a typo or my calculation is wrong. Wait the options: 6,8,5,3. Wait when spools left is 2, the comforters made: Let's see the line. At y = 2, x is 4.8, which is closest to 5? But wait the options: 6,8,5,3. Wait maybe I misread the graph. Wait the y - axis: spools of thread remaining. Let's check the options again. Wait the correct answer should be 6? No, wait when x = 6, y = 1.25; x = 5, y = 1.875; x = 4, y = 2.5. Oh! Wait, I think I made a mistake in the equation. Wait the line is from (0,5) to (8,0). So the slope is \(\frac{0 - 5}{8 - 0}=-\frac{5}{8}\), so the equation is \(y = 5-\frac{5}{8}x\). When y = 2, \(2=5-\frac{5}{8}x\), so \(\frac{5}{8}x=3\), \(x=\frac{24}{5}=4.8\approx5\)? But the options have 5. Wait the options: 6 comforters, 8 comforters, 5 comforters, 3 comforters. Wait maybe the graph is drawn with integer points. Let's look at the graph again. The line passes through (0,5) and (8,0). Let's list the points:
- When x = 0, y = 5
- x = 1, y = 4.375 (not integer)
- x = 2, y = 3.75 (not integer)
- x = 3, y = 3.125 (not integer)
- x = 4, y = 2.5 (not integer)
- x = 5, y = 1.875 (not integer)
- x = 6, y = 1.25 (not integer)
- x = 7, y = 0.625 (not integer)
- x = 8, y = 0

Wait, maybe the problem is that the line is a straight line and we can use the ratio. The total spools used: 5 - 2 = 3 spools used. Total spools available: 5 spools (at start). The number of comforters when 5 spools are used: 8 comforters. So the number of comforters when 3 spools are used: \(\frac{3}{5}\times8 = 4.8\approx5\). So the answer is 5 comforters? But wait the options: 6,8,5,3. So 5 comforters is an option.

Wait, maybe I made a mistake. Wait the graph: when spools left is 2, how many comforters? Let's check the options. Wait the correct answer is 6? No, wait when x = 6, y = 1.25; x = 5, y = 1.875; x = 4, y = 2.5. Oh! Wait, the y - axis is spools remaining. So when spools remaining is 2, the x (comforters) is 4.8, which is closest to 5. So the answer is 5 comforters? But let's check the options again. The options are 6,8,5,3. So 5 comforters is an option.

Wait, maybe the graph is drawn with each square as 1 unit, and the line is from (0,5) to (8,0). So the slope is - 5/8. So when y = 2, x=(5 - 2)*8/5=24/5 = 4.8≈5. So the answer is 5 comforters.

Answer:

5 comforters