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. t (years) v (dollars) 1 14 2 12 4 8 6 4 7 2 b. the t-intercept of the model is (8,0). the t-intercept estimates the year for which th stock will have no value. d. what is the v-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 v-intercept of the model is . the v-intercept estimates the value of the stock in 2000. b. the v-intercept of the model is . the v-intercept estimates the year for which the stock will have no value.

Explanation:

Step1: Find the linear model

First, we need to find the linear equation of the form \( v = mt + b \), where \( m \) is the slope and \( b \) is the \( v \)-intercept. Let's use two points, say \( (t_1, v_1)=(1, 14) \) and \( (t_2, v_2)=(2, 12) \). The slope \( m=\frac{v_2 - v_1}{t_2 - t_1}=\frac{12 - 14}{2 - 1}=\frac{- 2}{1}=-2 \).

Step2: Find the equation

Using the point - slope form \( v - v_1=m(t - t_1) \), with \( (t_1, v_1)=(1, 14) \) and \( m = - 2 \), we have \( v-14=-2(t - 1) \). Simplify it: \( v-14=-2t + 2 \), so \( v=-2t + 16 \).

Step3: Find the v - intercept

The \( v \)-intercept occurs when \( t = 0 \). Substitute \( t = 0 \) into the equation \( v=-2t + 16 \). We get \( v=-2(0)+16 = 16 \). So the \( v \)-intercept is the ordered pair \( (0,16) \). And the \( v \)-intercept estimates the value of the stock in 2000 (since \( t = 0 \) corresponds to the year 2000), so we choose option A.

Answer:

A. The v - intercept of the model is \((0,16)\). The v - intercept estimates the value of the stock in 2000.