Lindbergh's Air Speed Flying too fast or slow wastes fuel....

Lindbergh's Air Speed Flying too fast or slow wastes fuel. For the Spirit of St. Louis, miles per pound of fuel $M$ was a function of air speed $A$ in miles per hour, modeled by the formula $$ M=-0.000653 A^{2}+0.127 A-5.01 $$ a. Use the accompanying graph to estimate the most economical air speed. b. Use the formula to find the value of $A$ that maximizes $M$. c. How many gallons of fuel did he burn per hour if he flew at $97 \mathrm{mph}$ and got $1.2 \mathrm{mi} / \mathrm{lb}$ of fuel, which weighed $6.12 \mathrm{lb} / \mathrm{gal} ?$

Lindbergh's Air Speed Flying too fast or slow wastes fuel. For the Spirit of St. Louis, miles per pound of fuel $M$ was a function of air speed $A$ in miles per hour, modeled by the formula
$$
M=-0.000653 A^{2}+0.127 A-5.01
$$
a. Use the accompanying graph to estimate the most economical air speed.
b. Use the formula to find the value of $A$ that maximizes $M$.
c. How many gallons of fuel did he burn per hour if he flew at $97 \mathrm{mph}$ and got $1.2 \mathrm{mi} / \mathrm{lb}$ of fuel, which weighed $6.12 \mathrm{lb} / \mathrm{gal} ?$

Key Concepts

Unit Conversion and Dimensional Analysis

Unit conversion and dimensional analysis involve transforming one set of measurement units into another to maintain consistency and accuracy in calculations. This is critical in applied problems where various units are used, such as converting between fuel weight and volume, or translating miles per fuel weight into fuel consumption per time unit. These skills ensure that the computed results are practical and correctly interpreted in the real world.

Graphical Analysis

Graphical analysis refers to interpreting the plotted curve of a function to extract meaningful information, such as estimating the location of maximum or minimum values. In optimization problems, a graph can provide a visual understanding of where the function achieves its peak efficiency or performance, complementing analytical methods.

Quadratic Functions and Optimization

This concept involves understanding quadratic functions, particularly how their parabolic graphs behave. When a quadratic function has a negative leading coefficient, the parabola opens downward, meaning there is a maximum value at its vertex. Identifying this vertex either graphically or analytically (using the formula -b/(2a)) allows one to determine the maximum point of the function, which is essential in optimization problems.

Transcript

00:01 So for this question, the first thing they want us to do is to approximate from the graph the most economical speed.

00:09 Well, the graph that they give us, the speed is our horizontal axis, and miles per pound of fuel is our vertical axis.

00:32 Now, that means we want to maximize the vertical, or m, which is going to be our vertex right here.

00:41 Now, if we were to just drop straight down to see where this would be for a, well, that looks like it's about 95 miles per hour or so.

00:57 Now we want to find part a exactly, and well, that's the same.

01:05 The input of our vertex and we can find that by letting capital a equal to negative b over two times little a, or the little a and b are those coefficients for our quadrarcharch.

01:21 So plugging this in, we would get negative 0 .127 all over 2 times negative 0 .000653.

01:37 Now those negatives counsel out with each other and this is going to be exactly 63 ,500 over 653 miles per hour, which if we were to approximate this is going to be around 97 .2 miles per hour.

02:02 So it looks like our approximation from the graph is pretty good.

02:10 Now, we want to find the gallons of wool used if we're flying at 97 miles per hour, and that used 1 .2 miles per pound of fuel, which weighed 6 .12 pounds per gallon.

02:34 So what we want, so i'll write this off on the side here, so we want gallons per hour.

02:53 That we used.

02:56 Well first the only thing that we have for gallons is this right here.

03:02 So we want to reciprocate that first.

03:03 So let's go ahead and rewrite this as 1 over 6 .12.

03:11 So reciprocating that now gallons goes into the numerator and pounds is in the denominator.

03:16 And i'm going to use what is called a dimensional analysis chart to help us with this now.

03:22 Alright, so we want gallons in the numerator, hours in the denominator.

03:29 So we already have gallons in the numerator, so that's good.

03:34 But now we want to change pounds into hours.

03:36 So let's see what units we have.

03:39 Well, to get rid of the pounds, we can first use this 1 .2.