Question

a machine at a bakery is pouring out flour. the graph shows the amount of flour (in grams) in the machine versus time (in minutes).

amount of flour (grams)
time (minutes)

(a) what is the amount of flour in the machine at 0 minutes?

grams

(b) choose the statement that best describes how the time and amount of flour are related. then fill in the blank.

as time increases, the amount of flour in the machine

decreases.

at what rate is the amount of flour decreasing?

grams per minute

as time increases, the amount of flour in the machine

increases.

at what rate is the amount of flour increasing?

grams per minute

Explanation:

Step1: Analyze the graph for t=0

At \( t = 0 \) minutes (the y - intercept of the graph, since time is on the x - axis and amount of flour is on the y - axis), we look at the value of the amount of flour. From the graph, when \( t = 0 \), the amount of flour is the initial value. Looking at the graph, the y - intercept (when \( x = 0 \)) is 300 grams (assuming the graph's y - axis starts at 300 when \( x = 0 \), but wait, actually, looking at the graph, when time \( t = 3 \) minutes, the amount of flour is 0? Wait, no, maybe I misread. Wait, the x - axis is time (minutes) and y - axis is amount of flour (grams). Wait, the graph has a point at \( t = 3 \) minutes with amount 0? No, maybe the graph is of the machine pouring out flour, so at \( t = 0 \), the amount of flour in the machine is, say, 300 grams, and at \( t = 3 \) minutes, it's 0? Wait, no, the problem says "A machine at a bakery is pouring out flour. The graph shows the amount of flour (in grams) in the machine versus time (in minutes)". So when time \( t = 0 \), that's the initial amount. Let's re - examine the graph. If at \( t = 3 \) minutes, the amount of flour is 0, and at \( t = 0 \), let's see the y - axis. The y - axis has markings, maybe at \( t = 0 \), the amount is 300 grams? Wait, maybe the graph is a line from (0, 300) to (3, 0)? Wait, no, the blue line: let's check the coordinates. If the x - axis is time (minutes) with values 0,1,2,3,... and y - axis is amount (grams) with values 0,50,100,150,200,250,300,350,400,450. Wait, the blue line starts at (0, 300) and goes to (3, 0)? No, that would be decreasing. Wait, the problem part (b) has "As time increases, the amount of flour in the machine decreases" as an option. So at \( t = 0 \), the amount of flour is 300 grams. So for part (a), the amount of flour at 0 minutes is 300 grams.

Step2: Determine the rate of decrease

The rate of change (slope) of the line is \( \frac{\text{change in amount}}{\text{change in time}} \). The change in amount is \( 0 - 300=- 300 \) grams (since it goes from 300 to 0) and the change in time is \( 3 - 0 = 3 \) minutes. So the rate of decrease is \( \frac{-300}{3}=- 100 \) grams per minute. But since we are asked for the rate of decrease, it's 100 grams per minute. And the statement that describes the relationship is "As time increases, the amount of flour in the machine decreases".

Step3: For the increase rate (but since it's decreasing, maybe a mis - read, but if we consider the machine pouring out, it's decreasing. Wait, the last part: "At what rate is the amount of flour increasing?" But if the machine is pouring out, it should be decreasing. Maybe a typo, but following the graph, since it's a line from (0, 300) to (3, 0), the rate of decrease is 100 grams per minute, and the rate of increase would be 0 (but that doesn't make sense). Wait, no, the machine is pouring out, so the amount in the machine is decreasing. So for the rate of decrease, it's \( \frac{300 - 0}{3-0}=100 \) grams per minute (decreasing, so rate is - 100, but as a positive rate of decrease, 100 grams per minute).

Answer:

s:
(a) 300 grams
(b) Statement: "As time increases, the amount of flour in the machine decreases"; Rate of decrease: 100 grams per minute; Rate of increase: 0 (but since it's decreasing, the increase rate is not applicable, but if we consider the pouring out, the decrease rate is 100 grams per minute)

(Note: Since the problem has some ambiguity in the graph's exact coordinates, but based on the context of the machine pouring out flour, the initial amount at t = 0 is 300 grams, the relationship is decreasing, and the rate of decrease is 100 grams per minute.)