Question

the following table gives values for a function ( n = n(t) ).

( t )	( n = n(t) )
20	23.8
30	44.6
40	51.3
50	53.2
60	53.7
70	53.9

calculate the average rate of change from ( t = 30 ) to ( t = 40 ).

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use your answer to estimate the value of ( n(38) ). (round your answer to one decimal place.)

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Explanation:

Part 1: Average Rate of Change from \( t = 30 \) to \( t = 40 \)

Step 1: Recall the formula for average rate of change

The average rate of change of a function \( N(t) \) from \( t = a \) to \( t = b \) is given by \(\frac{N(b) - N(a)}{b - a}\). Here, \( a = 30 \), \( b = 40 \), \( N(30) = 44.6 \), and \( N(40) = 51.3 \).

Step 2: Substitute the values into the formula

\[
\frac{N(40) - N(30)}{40 - 30} = \frac{51.3 - 44.6}{10}
\]

Step 3: Calculate the numerator and then the fraction

First, \( 51.3 - 44.6 = 6.7 \). Then, \(\frac{6.7}{10} = 0.67\).

Step 1: Use the average rate of change to estimate

The average rate of change from \( t = 30 \) to \( t = 40 \) is \( 0.67 \) (per unit \( t \)). We want to find \( N(38) \), so we start from \( t = 30 \) and move \( 38 - 30 = 8 \) units of \( t \). The change in \( N \) is approximately the average rate of change times the change in \( t \), so \( \Delta N \approx 0.67 \times 8 \).

Step 2: Calculate the change in \( N \)

\[
0.67 \times 8 = 5.36
\]

Step 3: Add this change to \( N(30) \)

\( N(38) \approx N(30) + 5.36 = 44.6 + 5.36 = 49.96 \), which rounds to \( 50.0 \) (to one decimal place).

Answer:

\( 0.67 \)

Part 2: Estimate \( N(38) \)