Question

janell tutors math students and earns $20 an hour for each session. she spends about $60 a month on transportation and other costs related to tutoring. (a) write an equation for the monthly profit p in dollars that janell makes from tutoring for h hours. p = (b) find the slope of the equation in part (a). (c) find the vertical intercept of the equation. (h, p) = ( ) give its meaning in terms of her monthly profit. (d) find the horizontal intercept. (h, p) = ( ) give its meaning in terms of her monthly profit. in order to break even, janell must tutor for hours.

Explanation:

Step1: Write the profit - equation

The profit $P$ is the revenue minus the cost. The revenue from tutoring $h$ hours at $20$ dollars per hour is $20h$, and the cost is $60$. So the profit equation is $P = 20h-60$.

Step2: Find the slope

The equation $P = 20h - 60$ is in the form $y=mx + b$ (where $y = P$, $x = h$, $m$ is the slope and $b$ is the y - intercept). For the equation $P = 20h-60$, the slope $m = 20$. The slope represents the amount of additional profit per hour of tutoring.

Step3: Find the vertical intercept

The vertical - intercept (y - intercept) of the equation $P = 20h-60$ occurs when $h = 0$. When $h = 0$, $P=-60$. The meaning is that if Janell does not tutor any hours ($h = 0$), she will have a loss of $\$60$ (her transportation and other costs).

Step4: Find the horizontal intercept

The horizontal - intercept occurs when $P = 0$. Set $P = 0$ in the equation $P = 20h-60$. Then $0=20h - 60$. Solving for $h$:
\[

$$\begin{align*} 20h&=60\\ h& = 3 \end{align*}$$

\]
The horizontal intercept $(h,P)=(3,0)$. The meaning is that Janell must tutor for 3 hours to break even (her profit is $0$).

Answer:

(a) $P = 20h-60$
(b) Slope: 20 dollars per hour; it represents the increase in profit for each additional hour of tutoring.
(c) $(h,P)=(0, - 60)$; If Janell does not tutor any math students, she will lose 60 dollars.
(d) $(h,P)=(3,0)$; In order to break even, Janell must tutor for 3 hours.