Question

select the correct answer
an entrepreneur estimates his total profit (total revenue minus total cost) for his proposed company as $p(t)=t^3-4t^2+5t-20$, where $t$ is in hundreds of dollars and $t$ is the number of years the company has been in business. in which year ($t$) will the entrepreneur break even?
a. year 1
b. year 2
c. year 3
d. year 4

Explanation:

Step1: Set profit to 0 (break-even)

$P(t) = t^3 - 4t^2 + 5t - 20 = 0$

Step2: Factor by grouping

Group terms: $(t^3 - 4t^2) + (5t - 20) = 0$
Factor out common terms: $t^2(t - 4) + 5(t - 4) = 0$
Combine factors: $(t^2 + 5)(t - 4) = 0$

Step3: Solve for t

$t^2 + 5 = 0$ has no positive real solutions.
$t - 4 = 0 \implies t = 4$

Answer:

D. year 4