Question

penter at an rv factory uses nails in a pneumatic nailer at a rate of 20 nails per minute. there are 2500 nails in one box. the number of nails x minutes can be represented by a function. identify the graph that shows the number of nails remaining y as a function of time x, in minutes.
graph of the function is linear.
125
er than 0mber of minutes increases, the number of nails remaining decreases.

Explanation:

Step1: Define the function

The initial number of nails is 2500, and the rate of using nails is 20 per minute. So the function for the number of remaining nails \( y \) as a function of time \( x \) (in minutes) is \( y = 2500 - 20x \). This is a linear function in the form \( y=mx + b \), where \( m=- 20 \) (slope, negative because the number of nails is decreasing) and \( b = 2500 \) (y - intercept, initial number of nails).

Step2: Analyze the graph properties

• Y - intercept: When \( x = 0 \) (at the start, no time has passed), \( y=2500 \). So the graph should pass through the point \( (0,2500) \).
• Slope and trend: The slope is - 20, which means for every 1 - unit increase in \( x \) (1 minute), \( y \) decreases by 20. As \( x \) (number of minutes) increases, \( y \) (number of remaining nails) decreases. Also, the graph should be a straight line (since it's a linear function) with a negative slope.
• X - intercept: To find when the box is empty (\( y = 0 \)), we solve \( 0=2500 - 20x \). \( 20x=2500 \), so \( x=\frac{2500}{20}=125 \). So the graph should pass through \( (125,0) \).

So the correct graph should be a straight line starting at \( (0,2500) \), decreasing with a slope of - 20, and ending at \( (125,0) \) (or approaching \( y = 0 \) as \( x \) approaches 125, and \( x\geq0,y\geq0 \) since time and number of nails can't be negative).

Answer:

The graph should be a linear graph with a y - intercept at \( (0,2500) \), a negative slope (decreasing as \( x \) increases), and an x - intercept at \( (125,0) \). (If there are multiple graphs, the one that matches these properties: straight line, starts at (0,2500), decreases, and ends at or near (125,0) is the correct one.)