Question

savannah 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, what would you predict to be the length of a catfish that weighed 36 pounds?

Explanation:

Step1: Find the slope of the line

Use the formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(20,6)\) and \((x_2,y_2)=(23,12)\).
\(m=\frac{12 - 6}{23 - 20}=\frac{6}{3}=2\)

Step2: Find the equation of the line

Use the point - slope form \(y - y_1=m(x - x_1)\). Using the point \((20,6)\) and \(m = 2\):
\(y-6=2(x - 20)\)
\(y-6=2x-40\)
\(y = 2x- 34\)

Step3: Predict the length when \(y = 36\)

Substitute \(y = 36\) into the equation \(y = 2x-34\):
\(36=2x-34\)
Add \(34\) to both sides: \(36 + 34=2x\), \(70 = 2x\)
Divide both sides by \(2\): \(x=\frac{70}{2}=35\)

Answer:

The predicted length of the catfish is \(35\) inches.