Project 4. Literal Formula.
Most math books define the area of a circle as follows: \(A=\pi r^2\text{,}\) where A is the area of the circle and r is the radius of a circle. A text used in UAA’s Automotive Diesel program defines the area of a circle as \(A=0.7854d^2\text{,}\) where A is the area of the circle and d is the diameter of the circle.
The purpose of this project is to determine when each formula is most useful.
(a)
What is the mathematical relationship between radius and diameter? Your answer can be a sentence or an equation.
(b)
Show mathematically how to get from the formula \(A=\pi r^2\) to the formula \(A=0.7854d^2\text{.}\) This should take you multiple steps.
(c)
Explain in words what you did in each step to change the first formula into the second. What assumptions did you have to make? Anyone reading this answer should be able to replicate the math by just reading your answer. That is, talk me through all the steps.
(d)
Did you have any false starts or did you see how to change the formula right away? There is no wrong answer here; I just want you to think about your process.
(e)
For this problem, you will need a tape measure or a ruler. If doing this on a device it must be a computer and ensure you are at 100% magnification. Your phone or a scaled version will distort the results. First measure the radius of the circle in Figure 3.6.1. Then \(measure\) the diameter of the circle below and record your answer. Do not calculate the diameter! This must be measured, not calculated. Try to be as precise as is reasonably possible. Include units.
Was it easier to measure the radius or the diameter?
(f)
What is one reason why it might be more practical on a job to use the formula \(A=0.7854d^2\) instead of \(A=\pi r^2\text{?}\) If it helps, you may wish to ask yourself why the auto diesel students in particular use this less traditional formula.
(g)
Determine how many significant figures are in each measurement. If the number is not a measurement or the measurement has no error then it is called ‘exact’.
- \(\displaystyle \pi\)
- 0.7854
(h)
Which of the two formulas is more accurate? Which is more precise? Give a reason to back up your answer.
