Question

1. several students performed an experiment by dropping marbles of uniform size into a jar of water. they measured the height of the water after every five marbles. the results are listed in the table. which function best represents the problem situation and the approximate number of marbles it will take for the water to reach a height of 2 feet?

Explanation:

Step1: Find the slope

Use the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Take two points \((5,8)\) and \((10,10)\).
\(m=\frac{10 - 8}{10 - 5}=\frac{2}{5}=0.4\). Wait, let's check another pair. Take \((10,10)\) and \((15,11)\), \(m=\frac{11 - 10}{15 - 10}=\frac{1}{5}=0.2\). Wait, actually, if we use linear regression - like approach (or assume a linear model \(y=mx + b\)). Using the first two points \((x_1 = 5,y_1 = 8)\) and \((x_2=20,y_2 = 13)\)
\(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{13 - 8}{20 - 5}=\frac{5}{15}=\frac{1}{3}\approx0.3\)
Using the point - slope form \(y - y_1=m(x - x_1)\), with \(m = 0.3\) and \((x_1 = 5,y_1 = 8)\)
\(y-8=0.3(x - 5)\)
\(y-8=0.3x-1.5\)
\(y = 0.3x+6.5\)

Step2: Find the number of marbles for \(y = 20\)

Set \(y = 20\) in the equation \(y=0.3x + 6.5\)
\(20=0.3x+6.5\)
Subtract \(6.5\) from both sides: \(20 - 6.5=0.3x\)
\(13.5 = 0.3x\)
Divide both sides by \(0.3\): \(x=\frac{13.5}{0.3}=45\) (Wait, no. Wait, if the height \(y = 20\) (assuming a wrong initial thought. Wait, the problem says "height of 2 feet" (24 inches)).
Set \(y = 24\) (since height is in inches) in \(y = 0.3x+6.5\)
\(24=0.3x+6.5\)
\(24 - 6.5=0.3x\)
\(17.5=0.3x\)
\(x=\frac{17.5}{0.3}=\frac{175}{3}\approx58\)

Answer:

\(f(x)=0.3x + 6.5\); approximately 58 marbles