Question

let v be the value (in dollars) of a companys stock at t years since 2000. some pairs of values of t and v are shown in the following table. complete parts a to d.

b. estimate when the value of the stock was $6.
the value of the stock was $6 in 2005.
(round to the nearest year as needed.)

c. what is the t-intercept of the model? what does it mean in this situation?

select the correct choice and fill in the answer box to complete your choice.

(type an ordered pair.)

a. the t-intercept of the model is.
the t-intercept estimates the value of the stock in 2000.

b. the t-intercept of the model is.
the t-intercept estimates the year for which the stock will have no value.

t (years)	v (dollars)
2	12
4	8
6	4
7	2

Explanation:

Step1: Find the linear model

First, we find the slope \( m \) using two points. Let's take \((t_1, v_1)=(1, 14)\) and \((t_2, v_2)=(2, 12)\). The slope formula is \( m=\frac{v_2 - v_1}{t_2 - t_1}=\frac{12 - 14}{2 - 1}=\frac{- 2}{1}=-2\).
Using the point - slope form \( v - v_1=m(t - t_1) \), with \((t_1, v_1)=(1, 14)\) and \( m=-2 \), we get \( v-14=-2(t - 1) \).
Simplify the equation: \( v-14=-2t + 2 \), so \( v=-2t+16 \).

Step2: Find the t - intercept

The t - intercept occurs when \( v = 0 \). Set \( v = 0 \) in the equation \( v=-2t + 16 \).
We have the equation \( 0=-2t+16 \).
Add \( 2t \) to both sides: \( 2t=16 \).
Divide both sides by 2: \( t = 8 \).
The t - intercept is the point where \( v = 0 \), so the ordered pair is \((8,0)\).
The t - intercept represents the time (in years since 2000) when the stock value \( v = 0 \), which means it estimates the year for which the stock will have no value. So the correct option is B.

Answer:

B. The t - intercept of the model is \((8,0)\).
The t - intercept estimates the year for which the stock will have no value.