In each case below, the top vector is being subtracted from the bottom vector. All of the vectors have the same magnitude. Is the magnitude of the difference in the left case greater than, less than, or equal to the magnitude of the difference in the right case?
For each A*R*C*S activity, you are expected to complete and label all of the A*R*C*S Steps. For your first A*R*C*S activity, each step will walk you through the A*R*C*S process. A simple example is provided in A*R*C*S Example: Rectangular Park II.
A surveyor measures a hillside and describes the height of the hill \(y\) (in meters) as the following function of distance from the base of the hill \(x\) (also measured in meters):
His computer software reports that \(a_1 = 1.5 \times 10^{-5}\) and \(a_2 = 1.0 \times 10^{-10}\text{.}\) He has also identified three especially large trees (about \(30 \mathrm{~m}\) tall) whose bases are located at \(x_1 = 5 \mathrm{~m}\text{,}\)\(x_2 = 257 \mathrm{~m}\text{,}\) and \(x_3 = 300 \mathrm{~m}\text{.}\)
Identify known and unknown quantities with both a symbol and a number. In general, when a quantity is stated with a number of symbol, that is a known quantity.
Tip: It looks like the surveyor’s software did not report the specific units for the constants \(a_1\) and \(a_2\text{.}\) Determine appropriate units for these constants. Describe why these are sensible units given the equation for \(y(x)\text{.}\)
Compare your numerical answers to other numbers in the situation. Find at least one other distance to compare it to: should it be bigger or smaller than that other distance? Explain your reasoning.
What symbols does your numerical answer depend on? In what ways does it depend on each symbol? Why do you think this does (or does not) make physical sense, given the situation?
When you complete an A*R*C*S Activity, you are expected to carry out many different steps, some of which you might be able to skip sometimes and still arrive at a correct answer. Write a few sentences answering the following two questions:
