In life when we figure out processes at work or in science we often express the result in mathematical notation. This includes equations, functions, and other options. These are collectively known as models. They allow us to communicate what we know and calculate results as needed. To succeed in many jobs and to fully enjoy life we need to be proficient at reading and using models.
This section begins by presenting models and illustrates calculating some results from them. It progresses to solving equations (models) as a review of algebra skills. Finally we present tips on how to identify and use models arising in applications. These topics are continued with specific models in later sections.
Ohm’s Law relates three properties of electricity: voltage, current, and resistance. Voltage, measured in volts (V), is analagous to the amount of pressure to move the electrons. Current, measured in amperes (amps), is how much electricity is moving. Resistance measured in Ohms (\(\Omega\)), is, as it sounds, the resistance of a material to letting electricity flow.
We will consider how voltage must change as the needed current changes. These are measurements in a science model which means we will use significant digits rounding.
Lift is the force that keeps aircraft in the air.The lift equation explains factors that control the strength of lift produced by an airfoil (think wing or propellor). The factors included are air density, surface area of the airfoil, the coefficient of lift, and velocity. Air density is the amount of air per volume; you may see this as highs and lows on a weather map. It is also related to pressurizing aircraft flying at high altitude. The coefficient of lift incorporates multiple factors that are part of the design of the airfoil and how it is in use during flight.
We can consider the impact of changes in air density. Air density generally decreases as altitude increases. This is a science model with measurements, so we will use significant digits rounding.
Suppose the air density is 0.002378 slugs per cubic feet, surface area is 125 ft2, \(C_L=1.5617\text{,}\) and velocity is 84.4 ft⁄s. Calculate the lift.
Note in both forms of the law the units can be other than those listed (especially different scale like centimeters rather than meters). However, they must always match including the constant \(R\) which is looked up in reference books.
Suppose the initial conditions are \(P_1=101.3 \text{ Pa}\text{,}\)\(V_1=0.125 \text{ m}^3\text{,}\) and \(T_1=10.2^\circ \text{ C}\text{.}\) Also \(V_2=0.125 \text{ m}^3\) and \(T_2=50.7^\circ \text{ C}\text{.}\) Calculate the new pressure. This is a science model with measurements, so we will use significant digits rounding.
The previous section illustrated calculating model results without solving. This section presents additional example requiring limited solving and finishes with solving before any values have been substituted.
Recall the model for lift. Suppose we know the weight of the aircraft (\(w=239\bar{0}\) lbs), density of air (\(\rho=0.001869 \text{ slugs}/\text{ft}^3\)), wing surface area (\(s=165 \text{ ft}^2\)), and velocity (\(v=91.1 \text{ ft}/\text{sec}\)). Noting that lift must equal weight, what must the coefficient of lift be? This is a science model with measurements, so we will use significant digits rounding.
Suppose we know the initial conditions (\(P_1=1.00\) atm, \(V_1=1.35 \text{ ft}^3\text{,}\)\(T_1=51.2^\circ\) F) and also \(P_2=1.00\) atm and \(V_2=1.39 \text{ ft}^3\text{.}\) What must the new temperature (\(T_2\)) be? This is a science model with measurements, so we will use significant digits rounding.
The previous examples solved for a variable in a model after substituting numbers for the other variables. The next examples illustrate solving first. Note this process is the same as solving after substituting (same algebra) though there may be more steps. We might wish to solve this way, so it is easier to use the model multiple times.
Notice that the steps are the same algebra as if there were numbers. Also we could divide by \(v^2\) and we are not concerned with the square as part of solving for \(S\text{.}\)
Above we started with a model and were asked to do something with it. Normally we will start with a problem which does not identify a model to use. Very few problems you encounter in lifte come pre-labled with models. This section presents how to start with a problem and work to a solution by identifying the model first.
Next we write the mathematical model (equation or function).
Use the description to determine which application type (e.g., percent, proportion, linear model, etc.). Note units can suggest this (e.g., meters and meters squared indicate something was squared).
Finally we should check that our answer makes sense. We should not have negative prices (usually) or distances larger than the earth (when working with terrestrial problems).
You moved across town and rented a 20 foot moving truck for the day. You want to make sure the bill you received is correct. If you paid $81.03, for how many miles were you charged? Assume there were no extra fees.
We are given the price per mile ($0.79 per mile). There is also a fixed cost for the rental ($39.95). Adding the fixed cost and the milage cost will give us the total.
We know the total cost, which will leaves \(m\) in the equation, the number of miles, which is what we want to calculate. We can use the solving technique in Section 3.1
