Problem $2$

Math $1220-$ Project $2$

Related Rates

You Should Do This Problem On Your Own. Please do not work with other students on this problem.

I Will Help You! If you need help getting started or making progress on this problem, please ask me for advice! Once you have solved the problems, if you are unsure of how to write it up properly, please ask me for advice!

The Situation. A common style of car jack has the following basic design

The way it works is that when you turn the handle, the horizontal bolt turns, and its threads bring the left and right corners closer together, and this forces the top and bottom corners to move apart. For this particular jack,

the four circular joints each have radius $\frac{1}{2}$ inch

the flat plates on the top and the bottom are each $\frac{1}{8}$ inch thick

the centers of adjacent circular joints are $8$ inches apart from one another

one turn of the handle changes moves the centers of the left and right circular joints together, or apart, by $\frac{1}{16}$ inch together.

How to Get Full Credit.

You will need to solve two related rates problems. To get full credit you must solve them and present your answers with good communication: this means at a minimum that your work should be well organized and easy to read. Here are some things to consider when you come to your final writeup:

You will need to explain how to turn the real$-$world question into a math question about rates of change. This takes some real thinking for you to figure out, so it should take some real writing for you to explain it$.$

There are no variables in the given real$-$world question. If you use a variable, you should explain clearly what it is$.$

There no equations in the given real$-$world question. If you use an equation involving your variables, you should explain how you know that the equation is true $($for example 'Since these triangles are similar triangles...$\text{'}).$

All of the calculations you include in your solution should be well$-$organized and easy to follow.

Be sure to conclude by clearly stating your answer the given question in the same terms in which it is asked $($for example, the question does not have any variables, so if you answer it by saying $x=5,$ then you are not communicating well$).$

The Problems

Problem I At a certain moment, the top of the top plate of the jack is $6$ inches above the ground, and you are turning the handle at $20$ revolutions per second. How quickly is the height of the jack increasing at this moment?

Problem $2$ At a later moment, the top plate of the jack is $12$ inches above the ground, and you are turning the handle at $10$ revolutions per second. How quickly is the height of the jack increasing at this moment?

You may answer this problem by referring to your work in Problem I and explaining how the calculation would be different; but you must do so with good communication.