Question

functions and lines
interpreting the domain and range of a linear function in context
a machine at a bakery contains flour. when the machine is turned on, it begins to pour the flour out. the machine pours until it is empty, at which time it turns off.
let y be the amount of flour (in grams) in the machine, and let x be the amount of time (in minutes) from when the machine is turned on. the equation $y = -25x + 200$ models y as a function of x. as shown in the graph below, this model is accurate only from when the machine is turned on until it turns off. (the endpoints of the graph show when the machine is turned on and off.)
graph: x-axis is amount of time (minutes) from 0 to 10, y-axis is amount of flour in the machine (grams) from 0 to 500. a line starts at (0, 200) and ends at (8, 0).
use the graph to complete each part.
(a) the machine is turned on when the amount of time is \\(\square\\) minute(s). the machine stays on until the amount of time is \\(\square\\) minute(s).
each value in the select is an amount of time since the machine was turned on.
(b) the machine starts with \\(\square\\) gram(s) of flour in it. the machine continues to pour out flour until it has \\(\square\\) gram(s) of flour in it.
each value in the select is an amount of flour in the machine.

Explanation:

Part (a) Step 1: Determine machine on time

The machine is turned on at \( x = 0 \) (start time), and turns off when \( y = 0 \). Solve \( 0=-25x + 200 \):
\( 25x = 200 \) → \( x = 8 \). So on at \( 0 \) min, off at \( 8 \) min.

Part (a) Step 2: Identify domain

The domain is \( x \)-values (time since on), so "domain" is the set.

Part (b) Step 1: Initial flour amount

At \( x = 0 \), \( y=-25(0)+200 = 200 \) grams (start).

Part (b) Step 2: Final flour amount

When \( x = 8 \), \( y = 0 \) grams (empty).

Part (b) Step 3: Identify range

The range is \( y \)-values (flour amount), so "range" is the set.

Answer:

(a) The machine is turned on when the amount of time is \( \boldsymbol{0} \) minute(s). The machine stays on until the amount of time is \( \boldsymbol{8} \) minute(s). Each value in the \(\boldsymbol{\text{domain}}\) is an amount of time since the machine was turned on.
(b) The machine starts with \( \boldsymbol{200} \) gram(s) of flour in it. The machine continues to pour out flour until it has \( \boldsymbol{0} \) gram(s) of flour in it. Each value in the \(\boldsymbol{\text{range}}\) is an amount of flour in the machine.