Using Models Use the model given to answer the questions...

Using Models Use the model given to answer the questions about the object or process being modeled. An ocean diver models the pressure $P$ ( in $\mathrm{lb} / \mathrm{in}^{2} )$ ) at depth $d$ (in ft) by $$P=14.7+0.45 d$$ (a) Make a table that gives the pressure for each 10 -ft change in depth, from a depth of 0 ft to 60 $\mathrm{ft}$ . (b) If the pressure is $30 \mathrm{lb} / \mathrm{in}^{2},$ what is the depth?

Using Models Use the model given to answer the questions about the object or process being modeled.

An ocean diver models the pressure $P$ ( in $\mathrm{lb} / \mathrm{in}^{2} )$ )
at depth $d$ (in ft) by

$$P=14.7+0.45 d$$

(a) Make a table that gives the pressure for each 10 -ft
change in depth, from a depth of 0 ft to 60 $\mathrm{ft}$ .
(b) If the pressure is $30 \mathrm{lb} / \mathrm{in}^{2},$ what is the depth?

Key Concepts

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions to represent and analyze real-world phenomena. In this context, the diver's pressure while underwater is modeled using a linear equation, which facilitates understanding and prediction of physical behavior.

Linear Equations

Linear equations describe relationships with a constant rate of change between the independent and dependent variables. Here, the pressure increases proportionally with depth, making the equation P = 14.7 + 0.45d an ideal example of a linear relationship.

Dependent and Independent Variables

In the model, depth is the independent variable because it is the input value that determines the outcome, while pressure is the dependent variable as its value is dependent on the depth. Understanding this relationship is central to interpreting and using mathematical models.

Table Representation

Tabulating values is a useful technique for visualizing how the model behaves over a range of inputs. By creating a table for different depths and their corresponding pressures, one can clearly see the pattern implied by the linear equation.

Solving Equations

Solving equations involves determining unknown values from given information. In this case, when the pressure value is given, algebraic manipulation of the linear equation is used to find the corresponding depth, reinforcing skills in solving for a variable in a linear equation.

Transcript

00:01 Alright, so the first question is asking a making a table with, there's a change in depth from 0 feet to 60 feet, and we just have to find a pressure.

00:11 So the equation is p equals 14 .7 plus .45d, and while p stands for the pressure in pounds per inch square, the d sends for feet.

00:25 So what we're going to do is we're going to plug in zero first in d.

00:30 So pressure equals to 14 .7 plus 0 .45 times 0.

00:43 Then we're going to get 14 .7 plus 0 equals to 0.

00:50 And we're going to just say p1 for that one.

00:54 And pressure 2, i'm going to say 14 .7 plus 0 .4.

01:01 1 .45 times 10 because you have to realize that p and d, they are the variable.

01:13 So they're the one that changing it while the 14 .7 and .45 are the constants.

01:19 So they are the constant even though when the pressure and depth still changes.

01:25 And if you have more questions about those kind of concept, you can refer to page 160 on chapter 1, section 10, i can upload this screenshot for you right there.

01:42 So it says that whenever the k y equals kx and k is not zero, then we say that it's a direct variation.

01:51 So let's get going.

01:52 So p2 equals to 14 .7 plus 4 .5, and that will be equal to what? if you're pulling the calculator, you'll get 19 .2.

02:07 And p3, kip going on, 0 .45 times 20 this time, that will be 14 .7 plus 9, and that will be equal to 23 .7.

02:27 And pressure 4 equals to 14 .7 plus 0 .45 times 30.

02:36 That will be equal to 13 .5.

02:45 And if you're pulling the calculator, it will be 28 .2.

02:49 And you just keep going until we reach to 60.

02:54 So i'm going to not do the rest of them.

02:57 And you realize that there's a pattern starting to show up.

03:00 Oh, actually this is 14 .7...