Mastering Word Problems: A Key Skill for Success in Applied Mathematics

[mathdad]

I am reviewing angle of elevation and depression. I came across this "tricky" problem.

A car is traveling on a level road toward a mountain 2 Km high. The angle of elevation from the car to the top of the mountain changes from 6 degrees to 15 degrees. How far has the car traveled?

What is the set up for this application?
Can someone draw what the problem describes?
The change in terms of degrees is not too clear, honestly.

With a clear picture, I may be able to figure out the answer.

 

[MarkFL]

Here is a diagram:

\begin{tikzpicture}
\draw[blue,thick] (0,0) -- (30,0);
\draw[blue,thick] (30,0) -- (30,3.15);
\draw[blue,thick] (30,3.15) -- (0,0);
\draw[blue,thick] (18.24,0) -- (30,3.15);
\draw[gray,thin] (30,0.25) -- (29.75,0.25);
\draw[gray,thin] (29.75,0) -- (29.75,0.25);
\node[blue] at (31,1.59) {\large 2 km};
\node[blue] at (9.12,-0.5) {\large $x$};
\node[blue] at (24.12,-0.5) {\large $d$};
\node[blue] at (5,0.25) {$6^{\circ}$};
\node[blue] at (20.5,0.25) {$15^{\circ}$};
\end{tikzpicture}

Now, you may write, using the tangent function:

\(\displaystyle \tan\left(6^{\circ}\right)=\frac{2}{x+d}\)

\(\displaystyle \tan\left(15^{\circ}\right)=\frac{2}{d}\)

Finally, solve for $x$. :D

 

[mathdad]

Here is a diagram:

\begin{tikzpicture}
\draw[blue,thick] (0,0) -- (30,0);
\draw[blue,thick] (30,0) -- (30,3.15);
\draw[blue,thick] (30,3.15) -- (0,0);
\draw[blue,thick] (18.24,0) -- (30,3.15);
\draw[gray,thin] (30,0.25) -- (29.75,0.25);
\draw[gray,thin] (29.75,0) -- (29.75,0.25);
\node[blue] at (31,1.59) {\large 2 km};
\node[blue] at (9.12,-0.5) {\large $x$};
\node[blue] at (24.12,-0.5) {\large $d$};
\node[blue] at (5,0.25) {$6^{\circ}$};
\node[blue] at (20.5,0.25) {$15^{\circ}$};
\end{tikzpicture}

Now, you may write, using the tangent function:

\(\displaystyle \tan\left(6^{\circ}\right)=\frac{2}{x+d}\)

\(\displaystyle \tan\left(15^{\circ}\right)=\frac{2}{d}\)

Finally, solve for $x$. :D

Mark,

You are amazing. You remind me of a tutor I once communicated with named Soroban. I love how you easily took written information and created two trigonometric equations. Splending work! I hope to be able to do likewise in the near future.

 

[mathdad]

Here is a diagram:

\begin{tikzpicture}
\draw[blue,thick] (0,0) -- (30,0);
\draw[blue,thick] (30,0) -- (30,3.15);
\draw[blue,thick] (30,3.15) -- (0,0);
\draw[blue,thick] (18.24,0) -- (30,3.15);
\draw[gray,thin] (30,0.25) -- (29.75,0.25);
\draw[gray,thin] (29.75,0) -- (29.75,0.25);
\node[blue] at (31,1.59) {\large 2 km};
\node[blue] at (9.12,-0.5) {\large $x$};
\node[blue] at (24.12,-0.5) {\large $d$};
\node[blue] at (5,0.25) {$6^{\circ}$};
\node[blue] at (20.5,0.25) {$15^{\circ}$};
\end{tikzpicture}

Now, you may write, using the tangent function:

\(\displaystyle \tan\left(6^{\circ}\right)=\frac{2}{x+d}\)

\(\displaystyle \tan\left(15^{\circ}\right)=\frac{2}{d}\)

Finally, solve for $x$. :D

Mark,

You are amazing. You remind me of a tutor I communicated with back in 2006 named Soroban. I love how you easily took written information and created two trigonometric equations. Splending work! I hope to be able to do likewise in the near future. I can easily take it from here.

P.S. Why are you not getting paid as a tutor in your local area?

 

[MarkFL]

Usually, we are given the distance traveled, and the initial/final angles of inclination, and asked to find the height. In such a case, I would generalize:

\begin{tikzpicture}
\draw[blue,thick] (0,0) -- (30,0);
\draw[blue,thick] (30,0) -- (30,3.15);
\draw[blue,thick] (30,3.15) -- (0,0);
\draw[blue,thick] (18.24,0) -- (30,3.15);
\draw[gray,thin] (30,0.25) -- (29.75,0.25);
\draw[gray,thin] (29.75,0) -- (29.75,0.25);
\node[blue] at (30.5,1.59) {\large $h$};
\node[blue] at (9.12,-0.5) {\large $x$};
\node[blue] at (24.12,-0.5) {\large $d$};
\node[blue] at (5,0.25) {$\alpha$};
\node[blue] at (20.5,0.25) {$\beta$};
\end{tikzpicture}

Now we can write:

\(\displaystyle \tan(\alpha)=\frac{h}{x+d}\implies h=(x+d)\tan(\alpha)\)

\(\displaystyle \tan(\beta)=\frac{h}{d}\implies d=h\cot(\beta)\)

And so we have:

\(\displaystyle h=(x+h\cot(\beta))\tan(\alpha)\)

Solving for $h$, there results:

\(\displaystyle h=\frac{x}{\cot(\alpha)-\cot(\beta)}\)

This way we solve the problem only once, and obtain a formula we can then plug numbers into whenever we encounter a problem of this type. (Yes)

 

[MarkFL]

...P.S. Why are you not getting paid as a tutor in your local area?

I prefer to donate my time here (as many others here do), and at another site for vBulletin coding, to provide help free of charge. :D

 

[mathdad]

Usually, we are given the distance traveled, and the initial/final angles of inclination, and asked to find the height. In such a case, I would generalize:

\begin{tikzpicture}
\draw[blue,thick] (0,0) -- (30,0);
\draw[blue,thick] (30,0) -- (30,3.15);
\draw[blue,thick] (30,3.15) -- (0,0);
\draw[blue,thick] (18.24,0) -- (30,3.15);
\draw[gray,thin] (30,0.25) -- (29.75,0.25);
\draw[gray,thin] (29.75,0) -- (29.75,0.25);
\node[blue] at (30.5,1.59) {\large $h$};
\node[blue] at (9.12,-0.5) {\large $x$};
\node[blue] at (24.12,-0.5) {\large $d$};
\node[blue] at (5,0.25) {$\alpha$};
\node[blue] at (20.5,0.25) {$\beta$};
\end{tikzpicture}

Now we can write:

\(\displaystyle \tan(\alpha)=\frac{h}{x+d}\implies h=(x+d)\tan(\alpha)\)

\(\displaystyle \tan(\beta)=\frac{h}{d}\implies d=h\cot(\beta)\)

And so we have:

\(\displaystyle h=(x+h\cot(\beta))\tan(\alpha)\)

Solving for $h$, there results:

\(\displaystyle h=\frac{x}{\cot(\alpha)-\cot(\beta)}\)

This way we solve the problem only once, and obtain a formula we can then plug numbers into whenever we encounter a problem of this type. (Yes)

You are smart for sure. Do you have any idea how much money you can make in your local area as a high school math tutor? Tell me something, how do you keep from forgetting the early chapters in textbooks? For example, if I am reviewing precalculus, by the time I get to chapter 5, the first-four chapters are gone from memory. See my point?

What's the trick? I see that you answer questions in various forums. You know calculus as well as trig as well as geometry as well as algebra, etc. How do you keep all this material fresh in your mind?

- - - Updated - - -

I prefer to donate my time here (as many others here do), and at another site for vBulletin coding, to provide help free of charge. :D

Thank you for your contribution. If I had your knowledge of math, tutoring would be a part-time job for me at a reasonable and affordable rate per hour.

 

[MarkFL]

You are smart for sure. Do you have any idea how much money you can make in your local area as a high school math tutor? Tell me something, how do you keep from forgetting the early chapters in textbooks? For example, if I am reviewing precalculus, by the time I get to chapter 5, the first-four chapters are gone from memory. See my point?

What's the trick? I see that you answer questions in various forums. You know calculus as well as trig as well as geometry as well as algebra, etc. How do you keep all this material fresh in your mind?

The single most helpful thing that keeps things in my memory is to keep using them, which providing help here certainly does. I liken it to playing a musical instrument...as long as you keep taking the instrument out and playing it, you will improve, but leave the instrument in its case and you will get rusty.

I try to memorize as little as possible, and recall the techniques rather than the formulas. Of course some memorization is required, or even inevitable when you use a formula enough times, but what I mean is I don't set out to try to memorize a bunch of formulas. I think it is much more important to learn the concepts behind the formulas, the motivation and method for deriving them, etc. When you investigate how the formulas are derived, this will keep your algebra skills honed as well. Typically, the skills tend to build on themselves too...I was much better at algebra/trig. after taking calculus, for example. :D

 

[mathdad]

The single most helpful thing that keeps things in my memory is to keep using them, which providing help here certainly does. I liken it to playing a musical instrument...as long as you keep taking the instrument out and playing it, you will improve, but leave the instrument in its case and you will get rusty.

I try to memorize as little as possible, and recall the techniques rather than the formulas. Of course some memorization is required, or even inevitable when you use a formula enough times, but what I mean is I don't set out to try to memorize a bunch of formulas. I think it is much more important to learn the concepts behind the formulas, the motivation and method for deriving them, etc. When you investigate how the formulas are derived, this will keep your algebra skills honed as well. Typically, the skills tend to build on themselves too...I was much better at algebra/trig. after taking calculus, for example. :D

You are so right. I started playing guitar in the early 1970s. By the 1980s, I had become an intermediate player. In fact, I became the main guitar player at a local home Bible study group in Brooklyn, New York.

I learned basic guitar tablature and standard music notation. As the years went by, life got in the way. I now play guitar in my room at least 10 times per year very different from the 1970s and 1980s. I think I will start to play again on my days off beginning next week and not spend too much time with math all day long.
I love math but the guitar, especially hymns, are very important to me.

Lastly, I also feel that the public school system is cheating the students by not teaching proofs. Memorizing formulas does not make a mathematician. For example, direct and indirect geometric proofs have been banned from most NYC public high schools. No wonder students graduate and cannot spell geometry!

Questions:

1. Why didn't you become a math teacher?
2. What is the important math skill to master?
I say word problem solving. What do you say?
3. On youtube, a professor from Australia uploaded an entire course called Rational Trigonometry. I did not learn trigonometry "his way" back in high school over 30 years ago. The professor's name is NJ Wildberger. Watch a small segment of Rational Trigonometry. It is simply bizzare.

 

[MarkFL]

Questions:

1. Why didn't you become a math teacher?

Hmm...well, I simply didn't take that path in life. I got into computer programming back in the mid-80s because I found it interesting and fun at first, then I realized I could make money with it, and in that pursuit, I realized I needed to know some math. I've always had what people would call a good head for numbers, but unless you are a monumental genius, that will only take you so far. So, I went back to school primarily to learn math to be a better, more efficient, programmer. While I was employed by my school as a tutor, teaching wasn't something I ever considered as a career.

2. What is the important math skill to master? I say word problem solving. What do you say?

Well, the ability to apply mathematics to real world situations is a good skill to have, but I don't think I can put my finger on the single most overall important skill every budding mathematician should master. I'm really not mature enough as a mathematician to say, to be honest. I know enough to sometimes be helpful to those learning the basics, but that's about it.

In an applied field like physics or computer science, then certainly the ability to apply mathematics is crucial. Everyone who desires to take part in mathematics should learn the fundamentals, but then after that, the path you take depends largely on what you want to do with mathematics.

 

[mathdad]

Hmm...well, I simply didn't take that path in life. I got into computer programming back in the mid-80s because I found it interesting and fun at first, then I realized I could make money with it, and in that pursuit, I realized I needed to know some math. I've always had what people would call a good head for numbers, but unless you are a monumental genius, that will only take you so far. So, I went back to school primarily to learn math to be a better, more efficient, programmer. While I was employed by my school as a tutor, teaching wasn't something I ever considered as a career.
Well, the ability to apply mathematics to real world situations is a good skill to have, but I don't think I can put my finger on the single most overall important skill every budding mathematician should master. I'm really not mature enough as a mathematician to say, to be honest. I know enough to sometimes be helpful to those learning the basics, but that's about it.

In an applied field like physics or computer science, then certainly the ability to apply mathematics is crucial. Everyone who desires to take part in mathematics should learn the fundamentals, but then after that, the path you take depends largely on what you want to do with mathematics.

Thank you for your answers. At 52, I would be overjoyed to simply "master" word problems, if that's even possible. You see, what I do for a living has nothing to do with my two CUNY degrees and certainly nothing to do with mathematics.

In the past, I recall failing several exams, mainly word problems that, had I succeeded, my life would be in a better place today. It has been my experience that most exams leading to good jobs test each candidate's ability to reason his or her way to the answer. Employers are not interested in our ability to solve x + 5 = 10. Ok. Back to math, particularly precalculus. I am going to do my best to learn as much as possible in this website.