Question

1. at a school festival, a photo booth is selling photos. they collect $40 for every 16 photos they sell. which statement correctly identifies both the slope and its meaning in the situation? a. the slope is \LXI0, so the amount of money made for every photo is $0.40. c. the slope is \\(\frac{2}{5}\\), so to make $1, they have to sell 0.40 photos. d. the slope is \LXI2, so to make $1, they have to sell 2.5 photos. chart: money collected ($) on x - axis (0,20,40,60,80,100), number of photos sold on y - axis (0,8,16,24,32,40,48), with a line and plotted points

Explanation:

Step1: Calculate the slope

The slope \( m \) of a line is given by \( m=\frac{\text{change in } y}{\text{change in } x} \). Here, \( y \) is Money Collected (\$) and \( x \) is Number of Photos Sold. From the graph, when \( x = 16 \), \( y = 40 \). So slope \( m=\frac{40}{16}=\frac{5}{2} \)? Wait, no, wait: Wait, the axes: Number of Photos Sold is on the y - axis? Wait, no, looking at the graph: the vertical axis (y - axis) is Number of Photos Sold? Wait, no, the labels: "Money Collected (\$)" is horizontal (x - axis?), no, wait the graph: the bottom axis is Money Collected (\$) (from 0 to 100), and the left axis is Number of Photos Sold (from 0 to 48). Wait, so the line goes from (0,0) to (40,16)? Wait, no, the dot is at (20,8), (60,24), (80,32). Wait, let's take two points: (20,8) and (40,16). Then slope \( m=\frac{16 - 8}{40 - 20}=\frac{8}{20}=\frac{2}{5} \). Wait, that's the slope. Wait, slope is \( \frac{\text{change in number of photos}}{\text{change in money}} \)? Wait, no, the slope formula: if \( y \) is number of photos and \( x \) is money, then slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take (20,8) and (40,16): \( m=\frac{16 - 8}{40 - 20}=\frac{8}{20}=\frac{2}{5} \). Wait, but let's check the problem statement: they collect \$40 for every 16 photos. So the rate is \( \frac{40}{16}=\frac{5}{2}=2.5 \) dollars per photo? Wait, no, maybe I mixed up x and y. Let's re - define: Let \( x \) be number of photos, \( y \) be money collected. Then when \( x = 16 \), \( y = 40 \). So slope \( m=\frac{y}{x}=\frac{40}{16}=\frac{5}{2} \). Wait, now I'm confused. Wait the options: Option C says "The slope is \( \frac{2}{5} \), so to make \$1, they have to sell 0.40 photos." Wait, if slope is \( \frac{\text{photos}}{\text{money}} \), then slope \( m=\frac{\text{number of photos}}{\text{money}} \). So if \( x \) is money, \( y \) is photos, then slope \( m=\frac{y}{x} \). From (20,8): \( m=\frac{8}{20}=\frac{2}{5} \). So for \( x = 1 \) (money = \$1), \( y=\frac{2}{5}\times1 = 0.4 \) photos. Wait, but the problem says "they collect \$40 for every 16 photos", so \$40 per 16 photos, so per \$1, photos sold is \( \frac{16}{40}=0.4 \) photos. And the slope: if we model \( y=\text{photos} \), \( x=\text{money} \), then \( y = mx \), so \( m=\frac{\text{photos}}{\text{money}} \). So slope \( m=\frac{16}{40}=\frac{2}{5} \). So to make \$1, photos sold is \( \frac{2}{5}\times1 = 0.4 \) photos. So option C: "The slope is \( \frac{2}{5} \), so to make \$1, they have to sell 0.40 photos." Wait, but let's check the slope calculation again. Let's take two points: when money is \$20, photos are 8; when money is \$40, photos are 16. So slope \( m=\frac{16 - 8}{40 - 20}=\frac{8}{20}=\frac{2}{5} \). So the slope is \( \frac{2}{5} \), and the meaning: slope is \( \frac{\text{photos}}{\text{money}} \), so for each dollar, the number of photos sold is \( \frac{2}{5}=0.4 \) photos. So option C is correct? Wait, no, option C says "The slope is \( \frac{2}{5} \), so to make \$1, they have to sell 0.40 photos." Yes, because \( \frac{2}{5}\times1 = 0.4 \). Let's check other options: Option A: slope \( \frac{2}{5} \), money per photo: \( \frac{5}{2}=2.5 \) dollars per photo, but slope here is \( \frac{\text{money}}{\text{photo}} \)? No, if slope is \( \frac{\text{money}}{\text{photo}} \), then \( m=\frac{40}{16}=\frac{5}{2} \), but the option A says slope \( \frac{2}{5} \), which is wrong. Option B: slope \( \frac{5}{2} \), money per photo: \( \frac{5}{2}=2.5 \) dollars per photo, but slope \( \frac{5}{2} \) would be \( \frac{\text{money}}{\text{photo}} \), but the calculation from points: (16,40) and (8,20), slope \( \frac{40 - 20}{16 - 8}=\frac{20}{8}=\frac{5}{2} \). Wait, now I see the confusion: which variable is on which axis. If the x - axis is number of photos, and y - axis is money, then slope \( m=\frac{\text{money}}{\text{photo}}=\frac{40}{16}=\frac{5}{2} \). Then for each photo, money made is \( \frac{5}{2}=2.5 \) dollars. But the options: Option B says "The slope is \( \frac{5}{2} \), so the amount of money made for every photo is \$2.50." Wait, that's correct? Wait, let's re - examine the graph. The vertical axis: Number of Photos Sold (from 0 to 48), horizontal axis: Money Collected (\$) (from 0 to 100). So the line passes through (8,20), (16,40), (24,60), (32,80). Wait, so when number of photos (y) is 16, money (x) is 40. So if we model \( x = my \), where \( x \) is money, \( y \) is photos, then \( m=\frac{x}{y}=\frac{40}{16}=\frac{5}{2} \). So slope (if x is money, y is photos) is \( \frac{5}{2} \), meaning for each photo, money made is \( \frac{5}{2}=2.5 \) dollars. But the options: Option B: "The slope is \( \frac{5}{2} \), so the amount of money made for every photo is \$2.50." Wait, but earlier I thought the slope was \( \frac{2}{5} \), but that was when I mixed up x and y. Let's clarify:

- If we define the function as \( \text{Money} = f(\text{Photos}) \), then \( f(16)=40 \), so the slope (rate) is \( \frac{40}{16}=\frac{5}{2} \) dollars per photo. So for each photo, they make \( \frac{5}{2}=2.5 \) dollars. But the options:

Option A: slope \( \frac{2}{5} \), money per photo \( 0.4 \) – wrong.

Option B: slope \( \frac{5}{2} \), money per photo \( 2.5 \) – let's check the slope calculation. If \( y=\text{Money} \), \( x=\text{Photos} \), then slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{40 - 20}{16 - 8}=\frac{20}{8}=\frac{5}{2} \). So yes, slope is \( \frac{5}{2} \), and for each photo, money made is \( \frac{5}{2}=2.5 \) dollars. Wait, now I'm confused because earlier I thought the slope was \( \frac{2}{5} \), but that was when I had x and y reversed. Let's check the graph labels: "Money Collected (\$)" is on the horizontal axis (x - axis), "Number of Photos Sold" is on the vertical axis (y - axis). So the coordinates are (Money, Photos), so (20,8), (40,16), (60,24), (80,32). So the slope \( m=\frac{\text{Photos}_2-\text{Photos}_1}{\text{Money}_2 - \text{Money}_1}=\frac{16 - 8}{40 - 20}=\frac{8}{20}=\frac{2}{5} \). So slope is \( \frac{2}{5} \) (photos per dollar). So to make \$1, photos sold is \( \frac{2}{5}\times1 = 0.4 \) photos. So option C: "The slope is \( \frac{2}{5} \), so to make \$1, they have to sell 0.40 photos." is correct? Wait, but the problem statement says "they collect \$40 for every 16 photos", so \$40 per 16 photos, so per \$1, photos are \( \frac{16}{40}=0.4 \) photos, which matches option C. And the slope is \( \frac{16}{40}=\frac{2}{5} \), which is the slope (photos per dollar). So option C is correct. Wait, but let's check option B: "The slope is \( \frac{5}{2} \), so the amount of money made for every photo is \$2.50." If we calculate money per photo, it's \( \frac{40}{16}=2.5 \) dollars per photo, which is \( \frac{5}{2} \). But the slope here would be \( \frac{\text{Money}}{\text{Photos}}=\frac{40}{16}=\frac{5}{2} \), but in the graph, the x - axis is money, y - axis is photos, so slope is \( \frac{\text{Photos}}{\text{Money}}=\frac{2}{5} \). So there's a confusion in variable assignment. Let's re - express:

- Let \( x \) = Number of Photos, \( y \) = Money Collected. Then the point is (16,40), so the equation is \( y=\frac{40}{16}x=\frac{5}{2}x \). So slope is \( \frac{5}{2} \), and for each photo, money is \( \frac{5}{2}=2.5 \) dollars. But in the graph, the x - axis is money, y - axis is photos, so the graph is \( x=\frac{16}{40}y=\frac{2}{5}y \), so \( y=\frac{5}{2}x \) (if x is photos, y is money). So the graph is plotted with money on x - axis and photos on y - axis, so it's the inverse. So the slope of the line as plotted (money on x, photos on y) is \( \frac{\text{photos}}{\text{money}}=\frac{2}{5} \), but the rate of money per photo is \( \frac{5}{2} \). Now, the options:

Option A: "The slope is \( \frac{2}{5} \), so the amount of money made for every photo is \$0.40." No, \( \frac{2}{5} \) is photos per dollar, not money per photo.

Option B: "The slope is \( \frac{5}{2} \), so the amount of money made for every photo is \$2.50." If we consider the function \( y=\text{money} \), \( x=\text{photos} \), then slope is \( \frac{5}{2} \), and money per photo is \( 2.5 \), which is correct. But the graph is plotted with money on x and photos on y, so the slope of the graph (as drawn) is \( \frac{\text{photos}}{\text{money}}=\frac{2}{5} \), but the slope of the function \( y = f(x) \) where \( x \) is photos and \( y \) is money is \( \frac{5}{2} \).

Option C: "The slope is \( \frac{2}{5} \), so to make \$1, they have to sell 0.40 photos." Since to make \$1, photos needed are \( \frac{16}{40}=0.4 \), and the slope of the graph (money on x, photos on y) is \( \frac{2}{5} \) (photos per dollar), so this is correct.

Option D: "The slope is \( \frac{5}{2} \), so to make \$1, they have to sell 2.5 photos." No, 2.5 photos would give \( 2.5\times2.5 = 6.25 \) dollars, which is wrong.

Wait, now I'm really confused. Let's use the two - point formula on the graph. The graph has points (20,8) and (40,16) (money, photos). So slope \( m=\frac{16 - 8}{40 - 20}=\frac{8}{20}=\frac{2}{5} \). So the slope is \( \frac{2}{5} \) (photos per dollar). So for each dollar (change in x by 1), change in y (photos) is \( \frac{2}{5}=0.4 \). So to make \$1 (x = 1), y (photos) is 0.4. So option C is correct.

Step2: Analyze each option

- Option A: Slope \( \frac{2}{5} \), but money per photo would be \( \frac{5}{2}=2.5 \), not 0.4. So A is wrong.
- Option B: Slope \( \frac{5}{2} \), but the slope of the graph (as plotted) is \( \frac{2}{5} \) (since x is money, y is photos). So B is wrong.
- Option C: Slope \( \frac{2}{5} \), and to make \$1, photos sold is \( \frac{2}{5}\times1 = 0.4 \), which matches \( \frac{16}{40}=0.4 \). So C is correct.
- Option D: Slope \( \frac{5}{2} \), and 2.5 photos would make \( 2.5\times2.5 = 6.25 \) dollars, not \$1. So D is wrong.

Answer:

C. The slope is \(\frac{2}{5}\), so to make \$1, they have to sell 0.40 photos.