Question

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

t	n = n(t)
20	25.5
30	49.9
40	52.5
50	54.2
60	54.7
70	54.9

calculate the average rate of change from t = 10 to t = 20.

use your answer to estimate the value of n(14). (round your answer to one decimal place.)

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Explanation:

Part 1: Average Rate of Change from \( t = 10 \) to \( t = 20 \)

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, \( a = 10 \), \( b = 20 \), \( N(10) = 16.8 \), and \( N(20) = 25.5 \).

Step 2: Substitute the values into the formula

\[
\text{Average rate of change} = \frac{N(20) - N(10)}{20 - 10} = \frac{25.5 - 16.8}{10}
\]

Step 3: Calculate the numerator and then the fraction

First, \( 25.5 - 16.8 = 8.7 \). Then, \( \frac{8.7}{10} = 0.87 \).

Step 1: Use the average rate of change to estimate

The average rate of change from \( t = 10 \) to \( t = 20 \) is \( 0.87 \), which means the function \( N(t) \) changes at an average rate of \( 0.87 \) per unit \( t \). We can use the linear approximation: \( N(t) \approx N(10) + \text{average rate of change} \times (t - 10) \) for \( t \) near 10. Here, \( t = 14 \), so \( t - 10 = 4 \).

Step 2: Substitute the values into the approximation formula

\[
N(14) \approx N(10) + 0.87 \times (14 - 10) = 16.8 + 0.87 \times 4
\]

Step 3: Calculate the product and then the sum

First, \( 0.87 \times 4 = 3.48 \). Then, \( 16.8 + 3.48 = 20.28 \). Rounding to one decimal place, we get \( 20.3 \).

Answer:

\( 0.87 \)

Part 2: Estimate \( N(14) \)