Question: Bao decides to research the relationship between the length in inches and the weight of a certain species of catfish. She measures the length and weight of a number of specimens she catches, then throws back into the water. After plotting all her data, she draws a line of best fit. Based on the line of best fit, how much would you predict a catfish with a length of 32 inches would weigh?
Catfish Weight (pounds)
37
34
31
28
25
22
19
16
13
10
7
4
y
(20,4)
(23,10)
(26,16)
(29,22)
x
Catfish Length (inches)
The graph shows a scatter plot with an upward-sloping line of best fit passing through the labeled points (20,4), (23,10), (26,16), and (29,22); the plotted trend suggests that at x = 32 inches the predicted weight is about 28 pounds.
1. **State the problem:** We want to predict the weight $y$ of a catfish given its length $x=32$ inches using the line of best fit from the data points.
2. **Identify the points on the line of best fit:** The points given are $(20,4)$, $(23,10)$, $(26,16)$, and $(29,22)$.
3. **Find the slope $m$ of the line:** Use two points, for example $(20,4)$ and $(29,22)$.
$$m = \frac{22 - 4}{29 - 20} = \frac{18}{9} = 2$$
4. **Find the equation of the line:** Use point-slope form with point $(20,4)$:
$$y - 4 = 2(x - 20)$$
Simplify:
$$y = 2x - 40 + 4 = 2x - 36$$
5. **Predict the weight for $x=32$ inches:** Substitute $x=32$ into the equation:
$$y = 2(32) - 36 = 64 - 36 = 28$$
6. **Interpretation:** The predicted weight of a catfish with length 32 inches is $28$ pounds.
**Final answer:** $28$ pounds.