Question

• for what values of a do you expect s(a) to be positive? why? • for what values of a do you expect s(a) to be negative? why? • for what values of a do you expect s(a) to be zero? why? (9) a rapidly growing city in arizona has its population p at time t, where is the number of decades after the year 2010 modeled by the formula p(t) = 25000e^{t/5}. use this function to respond to the following questions. • sketch an accurate graph of p for t = 0 to t = 5 and label your axes. • compute the average rate of change of p between 2030 and 2050. include units on your answer and give a one sentence explanation of this value in context. • use the limit definition to write an expression for the instantaneous rate of change of p with respect to time, t, at the instant a = 2. explain why this limit is difficult to evaluate exactly. • estimate the limit above for the instantaneous rate of change of p at the instant a = 2 by using several small values for h. once you have a reasonable estimate of p(2) include units in your answer and write a sentence (using everyday language) to explain the meaning of this value. • on your graph sketch two lines, one whose slope represents the average rate of change of p from t = 2 to t = 4 and the other whose slope represents the instantaneous rate of change of p at the instant a = 2. • in a carefully worded sentence, describe the behavior of p(a) as a increases in value. what does this reflect about the behavior of the given function p? (10) one of the longest stretches of straight (and flat) road in north america can be found in north dakota on state highway 46, which lies just south of the interstate highway i-94 and runs through the town of gackle. a car leaves town (at time t = 0) and heads east on highway 46; its poistion in miles from gackle at time t in minutes is given by the following graph- • in everyday language, describe the behavior of the car over the provided time interval. in particular, discuss what is happening on the time intervals t = 57 to t = 68 and t = 68 to t = 104. item find the slope of the line between the points (57, 63.8) and (104, 106.8). what are the units on this slope? what does this slope represent?

Explanation:

To solve the problem of finding the slope between the points \((57, 63.8)\) and \((104, 106.8)\) (from the car's position problem):

Step 1: Recall the slope formula

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Here, \(x_1 = 57\), \(y_1 = 63.8\), \(x_2 = 104\), and \(y_2 = 106.8\).

Step 2: Substitute the values into the formula

\[

$$\begin{align*} m&=\frac{106.8 - 63.8}{104 - 57}\\ &=\frac{43}{47}\\ &\approx0.9149 \end{align*}$$

\]

The units of the slope: The \(y\)-axis is in miles (position) and the \(x\)-axis is in minutes (time), so the units of the slope are miles per minute. This slope represents the average rate of change of the car's position with respect to time, i.e., the average speed of the car between \(t = 57\) minutes and \(t = 104\) minutes.

Answer:

The slope is \(\frac{43}{47}\approx0.91\) miles per minute. It represents the car's average speed between \(t = 57\) and \(t = 104\) minutes.