Question

liam wants to start up his own garage. he gets a small business loan to cover costs, and the value of the garage (in thousands of dollars) t months after its creation is modeled by: $v(t)=2t^{2}-24t - 40$
identify the variables.
x - axis: months and y - axis: thousand dollars
what is the y - intercept and what does it mean in context?
$(0, - 40)$ his company has cost $40,000 to start - up
approximate the lowest net value for liams garage? when did this happen?
the lowest net value of - 104 dollars happens at 40 months
approximate when liams company will start being worth money.

Explanation:

Step1: Encontrar el valor mínimo de la función \(v(t) = 2t^{2}-24t - 40\)

La función \(v(t)\) es una parábola de la forma \(y = ax^{2}+bx + c\), donde \(a = 2\), \(b=-24\), \(c = -40\). El vértice de una parábola \(y = ax^{2}+bx + c\) se da por \(t=-\frac{b}{2a}\)

$$t=-\frac{-24}{2\times2}=6$$

Step2: Calcular el valor de \(v(t)\) en \(t = 6\)

Sustituir \(t = 6\) en la función \(v(t)\)

$$v(6)=2\times6^{2}-24\times6 - 40=2\times36-144 - 40=72-144 - 40=-112$$

Como \(y\) está en miles de dólares, el valor mínimo es \(- 112000\) dólares y ocurre en \(t = 6\) meses.

Answer:

El valor mínimo neto es de \(-112000\) dólares y ocurre a los \(6\) meses.