Question

problem set

1. **a local grocery delivery service charges a flat pickup fee plus a fixed amount per mile. on tuesday a customer is charged $12.50 for a 5-mile delivery. the per-mile rate is $1.80.

(a) on the coordinate grid shown, plot the point corresponding to the 5-mile delivery. use the information given to graph the rest of the line. label the y-intercept and the point you plotted.
(b) write a linear equation for the total charge t (in dollars) as a function of distance x (in miles).
(c) explain the meaning of the slope in this situation.
(d) explain the meaning of the y-intercept in this situation.
graph: delivery charge vs. distance
graph with x-axis: distance x (miles) from 0 to 12, y-axis: total charge t (dollars) from 0 to 25

1. write a linear equation in slope-intercept form to model each situation.

(a) you rent a bicycle for $20 plus $2 per hour.
(b) an auto repair shop charges $50 plus $25 per hour.
(c) a candle is 6 inches tall and burns at a rate of \\(\frac{1}{2}\\) inch per hour.
(d) the temperature is 15° and is expected to fall 2° each hour during the night.
© 2025 uncommon schools, inc. all rights reserved.

Explanation:

Part (1b)

Step1: Recall linear equation form

The slope - intercept form of a linear equation is $T = mx + b$, where $m$ is the slope (rate of change) and $b$ is the y - intercept (initial value). We know the per - mile rate (slope $m$) is $\$1.80$ per mile, so $m = 1.8$. We also know that when $x = 5$ (miles), $T=12.50$ (dollars). We can use the point - slope form $T - T_1=m(x - x_1)$ or substitute into the slope - intercept form to find $b$.
Substitute $x = 5$, $T = 12.50$ and $m = 1.8$ into $T=mx + b$:
$12.50=1.8\times5 + b$

Step2: Solve for $b$

First, calculate $1.8\times5=9$. Then the equation becomes $12.50 = 9 + b$. Subtract 9 from both sides: $b=12.50 - 9=3.5$.
So the linear equation is $T = 1.8x+3.5$.

Step1: Identify slope and y - intercept

The slope - intercept form is $y=mx + b$, where $y$ is the total cost, $m$ is the rate per hour, and $b$ is the initial cost. For renting a bicycle, the initial cost (flat fee) $b = 20$ dollars, and the rate per hour $m = 2$ dollars per hour. Let $y$ be the total cost and $x$ be the number of hours.

Step2: Write the equation

Using the slope - intercept form $y=mx + b$, we substitute $m = 2$ and $b = 20$. So the equation is $y = 2x+20$.

Step1: Identify slope and y - intercept

The slope - intercept form is $y=mx + b$, where $y$ is the total cost of auto repair, $m$ is the rate per hour, and $b$ is the initial cost. The initial cost (flat fee) $b = 50$ dollars, and the rate per hour $m = 25$ dollars per hour. Let $y$ be the total cost and $x$ be the number of hours.

Step2: Write the equation

Using the slope - intercept form $y=mx + b$, we substitute $m = 25$ and $b = 50$. So the equation is $y=25x + 50$.

Answer:

$T = 1.8x + 3.5$

Part (2a)