A mathematical model for Merck's sales is given by...

A mathematical model for Merck's sales is given by $$ S(t)=-0.18 t^{2}+2.5 t+14 $$ where $t$ is time in years and $t=0$ corresponds to 1997. (A) Compare the model and the data graphically and numerically. (B) Estimate (to two significant digits) Merck's sales in 2006 and in 2008. (C) Write a brief verbal description of Merck's sales from 1997 to 2005.

A mathematical model for Merck's sales is given by
$$
S(t)=-0.18 t^{2}+2.5 t+14
$$
where $t$ is time in years and $t=0$ corresponds to 1997.
(A) Compare the model and the data graphically and numerically.
(B) Estimate (to two significant digits) Merck's sales in 2006 and in 2008.
(C) Write a brief verbal description of Merck's sales from 1997
to 2005.

Key Concepts

Verbal Interpretation

Verbal interpretation is about transforming numerical and graphical findings into descriptive language. This process helps in communicating the observed trends and behaviors in the model, providing a narrative that summarizes how the subject, such as sales, evolved over a certain period.

Numerical Estimation

Numerical estimation involves calculating approximate values using the mathematical model by substituting specific input values. This technique is essential for making concrete predictions and comparing them with observed data, especially when precise values are needed for future projections.

Mathematical Modeling

Mathematical modeling involves representing real-world phenomena using mathematical expressions. In this context, a function is used to approximate sales over time, encapsulating the trends observed in the data with a formula that can be manipulated, analyzed, and used for prediction.

Quadratic Functions

Quadratic functions, represented by expressions in the form ax² + bx + c, are used to describe relationships that exhibit a parabolic trend. Their structure allows for the analysis of turning points and the rate of change, which is particularly useful in modeling scenarios where growth accelerates or decelerates over time.

Graphical Comparison

Graphical comparison entails plotting both the model's predictions and the actual data points on a graph to visually assess how well the model fits the data. This approach helps in identifying trends, discrepancies, and overall validity of the mathematical representation compared to empirical observations.

Transcript

00:01 All right, welcome back.

00:02 We're going to be looking at merck sales here.

00:04 So sales function s of t to go to negative.

00:09 1 .182 plus 2 .5t plus 14.

00:16 So if we have the years 97, 99, 01, 03, and we're going to estimate 06 and 07, and the actual values of the sales are 14 .0, 17 .3, 21 .2, 22 .5, 22 .0.

00:52 And our estimate is, well, if we plug in zero into the equation, oops, it's supposed to be a 14.

01:01 We'll get 14.

01:03 Then if we plug in two, right, two years later, we'll get 18, point two eight plug in four and you'll get 21 .1 two plug in six and you'll get 22 .5 to and then plug in eight and you'll get 22 .48 again then kind of extrapolate right and go outside of our known values and plug in nine and that will give us a value of 21 .92.

01:43 Looks like merck isn't doing quite as well.

01:46 And then if you plug in 10, you'll get 21 .0.

01:52 It says to round two significant digits, so we'll round that 21 and so we're around to 22.

02:01 All right, we can draw a graph of this thing.

02:08 All right.

02:16 I'll draw a graph right there.

02:27 So this graph, this, these, uh, let's say what was the of merck.

02:34 Let's see, let's say he makes 14 this first year.

02:40 We're going up by 2s, 1, 2, 3, 4, 5, 6, 7.

02:45 Actually, that's the one that won't fit...