The following table shows the percentage of the U.S. Discretionary Budget allocated to education in 2003, 2005, and 2009 (t=0 represents 2000)

Year t 3 5 9
Percentage p(t) 6.8 7 6.2

(a) Compute and interpret the average rate of change of P(t)
(b) Over the period 2003-2009 (that is [3,9]) and
(c) Over the period [5,9] be sure to state the units of measurement.
I have no idea how to start this could you please walk me though how to work this problem? I am so lost. Thank you.

a has to be instructions for b, and c.

b. rate change=change/time=(6.2-6.8)/(9-3)

c. rate change=change/time=(7-6.8)/(9-5)

To compute the average rate of change of P(t), we need to find the difference in the values of P(t) over the given time period divided by the difference in the corresponding time values.

(a) To find the average rate of change of P(t) over the period [3, 9]:

Step 1: Calculate the difference in P(t):
Difference in P(t) = p(9) - p(3)
= 6.2 - 6.8
= -0.6

Step 2: Calculate the difference in time values:
Difference in t = 9 - 3 = 6

Step 3: Calculate the average rate of change:
Average rate of change = Difference in P(t) / Difference in t
= -0.6 / 6
= -0.1

Interpretation: The average rate of change of P(t) over the period [3, 9] is -0.1. This means that education allocation in the U.S. discretionary budget decreased by 0.1% per year on average during this period.

(b) To find the average rate of change of P(t) over the period [5, 9]:

Step 1: Calculate the difference in P(t):
Difference in P(t) = p(9) - p(5)
= 6.2 - 7
= -0.8

Step 2: Calculate the difference in time values:
Difference in t = 9 - 5 = 4

Step 3: Calculate the average rate of change:
Average rate of change = Difference in P(t) / Difference in t
= -0.8 / 4
= -0.2

Interpretation: The average rate of change of P(t) over the period [5, 9] is -0.2. This means that education allocation in the U.S. discretionary budget decreased by 0.2% per year on average during this period.

The units of measurement for the average rate of change are percentage points per year (%/year).

Sure, let's walk through each part of the problem step by step.

(a) To compute the average rate of change of P(t) over the entire time period [3,9], you need to find the change in the value of P(t) divided by the change in time. In this case, the change in P(t) is the difference between the value of P at time t=9 and the value of P at time t=3, and the change in time is 9-3=6.

So, the average rate of change is given by:
Average rate of change = (P(9) - P(3))/(9-3)

Plugging in the values from the table:
Average rate of change = (6.2 - 6.8)/(9-3)
Average rate of change = (-0.6)/6
Average rate of change = -0.1

Interpretation: The average rate of change of P(t) over the period from 2003 to 2009 is -0.1. This means that, on average, the percentage of the discretionary budget allocated to education decreased by 0.1% per year during this period.

(b) To compute the average rate of change of P(t) over the period [3,9] (2003-2009), follow the same steps as in part (a). The only difference is that the change in time is now 9-3 = 6 years.

Average rate of change = (P(9) - P(3))/(9-3)

Plugging in the values from the table:
Average rate of change = (6.2 - 6.8)/(9-3)
Average rate of change = (-0.6)/6
Average rate of change = -0.1

Interpretation: The average rate of change of P(t) over the period from 2003 to 2009 is -0.1. The units of measurement in this case are percentage points per year.

(c) To compute the average rate of change of P(t) over the period [5,9] (2005-2009), the process is the same as in part (a) and (b), but now the change in time is 9-5 = 4 years.

Average rate of change = (P(9) - P(5))/(9-5)

Plugging in the values from the table:
Average rate of change = (6.2 - 7)/(9-5)
Average rate of change = (-0.8)/4
Average rate of change = -0.2

Interpretation: The average rate of change of P(t) over the period from 2005 to 2009 is -0.2. The units of measurement in this case are percentage points per year.

I hope this explanation helps! Let me know if you have any further questions.