Question

math 98 2.2 worksheet

1. similar to wiley #21 a company’s profit after t months of operation is given by p(t)=1000 + 500(t - 4).

a) what is the practical meaning of the constants 4 and 1000? use proper units.
b) rewrite the function in slope - intercept form (that is, in the form p(t)=b+mt) and give a practical interpretation of each constant. use proper units in your interpretation.

Explanation:

Step1: Analyze the constant 4 in $P(t)$

The term $(t - 4)$ in the profit function indicates that the linear - growth part of the profit starts at $t = 4$ months.

Step2: Analyze the constant 1000 in $P(t)$

When $t = 4$, $P(4)=1000$, so 1000 is the profit at $t = 4$ months.

Step3: Rewrite the function in slope - intercept form

Expand $P(t)=1000 + 500(t - 4)$ using the distributive property: $1000+500t-2000=500t - 1000$.

Step4: Interpret the slope $m$

The slope $m = 500$ in $P(t)=500t - 1000$ represents the monthly increase in profit.

Step5: Interpret the y - intercept $b$

The y - intercept $b=-1000$ represents the profit at $t = 0$ months.

Answer:

a) The constant 4 in the function $P(t)=1000 + 500(t - 4)$ represents the number of months. It is the number of months after which the linear - growth part of the profit function starts. That is, the company's profit does not follow the linear pattern $500(t - 4)$ until $t = 4$ months. The constant 1000 represents the profit (in dollars) of the company at $t = 4$ months.
b) First, expand the function:
\[

$$\begin{align*} P(t)&=1000+500(t - 4)\\ &=1000 + 500t-2000\\ &=500t - 1000 \end{align*}$$

\]
In the slope - intercept form $P(t)=b+mt$, $m = 500$ and $b=- 1000$. The slope $m = 500$ represents the rate of change of the profit with respect to time. The unit is dollars per month, meaning the company's profit increases by 500 dollars per month after the first 4 months. The constant $b=-1000$ represents the profit (in dollars) of the company at $t = 0$ months. In a practical sense, it means the company had a debt or a negative profit of 1000 dollars at the start of its operation.