Question

current attempt in progress
a companys total profit after t months of operation is given by p(t)=1500 + 900(t - 7).
(a) what is the practical meaning of the constants 7 and 1500?
(b) rewrite the function in slope - intercept form and give a practical interpretation of the constants.
p(t)=i+it
the constant term is
the coefficient of t is
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hint

Explanation:

Step1: Interpret constants in original function

The number 7 and 1500 are analyzed based on business - profit context.

Step2: Expand the function

\[P(t)=1500 + 900t-6300\]

Step3: Simplify to slope - intercept form

\[P(t)=-4800 + 900t\]

Step4: Interpret new constants

The constant \(-4800\) and coefficient 900 are related to profit - loss and profit - rate respectively.

Answer:

(a) The constant 7 likely represents the number of months after which a certain change in the profit - calculation pattern occurs (perhaps the start of a new business phase or a change in cost - revenue structure). The constant 1500 might be an initial profit amount or a base profit value before considering the effect of the term related to \(t - 7\).
(b) First, expand the function \(P(t)=1500 + 900(t - 7)\):
\[

$$\begin{align*} P(t)&=1500+900t-6300\\ P(t)&=- 4800 + 900t \end{align*}$$

\]
So \(P(t)=-4800 + 900t\). The constant term \(-4800\) could represent an initial debt or a negative starting - point for profit (a loss that the company is trying to recover from). The coefficient of \(t\), which is 900, represents the rate of change of the profit with respect to time. That is, the company's profit increases by 900 units per month.