To find the average rate of change of the functions over the interval \([1.1, 1.6]\), we use the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \(a = 1.1\) and \(b = 1.6\).
- For the function \(y = 16 \cdot 4^x\):
First, we need to evaluate it at \(x = 1.1\) and \(x = 1.6\):
- \(f(1.1) = 16 \cdot 4^{1.1}\)
- \(f(1.6) = 16 \cdot 4^{1.6}\)
Calculating \(4^{1.1}\) and \(4^{1.6}\):
- \(4^{1.1} \approx 4.57\)
- \(4^{1.6} \approx 6.49\)
Now,
- \(f(1.1) \approx 16 \cdot 4.57 \approx 73.12\)
- \(f(1.6) \approx 16 \cdot 6.49 \approx 103.84\)
Now, calculate the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(1.6) - f(1.1)}{1.6 - 1.1} = \frac{103.84 - 73.12}{0.5} = \frac{30.72}{0.5} = 61.44 \]
- For the function \(y = 16x^2 + 4x\):
Next, we evaluate this at \(x = 1.1\) and \(x = 1.6\):
- \(f(1.1) = 16(1.1^2) + 4(1.1)\)
- \(f(1.6) = 16(1.6^2) + 4(1.6)\)
Calculating \(f(1.1)\) and \(f(1.6)\):
- \(f(1.1) = 16(1.21) + 4(1.1) = 19.36 + 4.4 = 23.76\)
- \(f(1.6) = 16(2.56) + 4(1.6) = 40.96 + 6.4 = 47.36\)
Now, calculate the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(1.6) - f(1.1)}{1.6 - 1.1} = \frac{47.36 - 23.76}{0.5} = \frac{23.6}{0.5} = 47.2 \]
- Comparing the average rates of change:
Now we need to find how much greater \(61.44\) is than \(47.2\):
\[ 61.44 - 47.2 = 14.24 \]
Finally, we can determine how much greater this is than the options given (1, 10, or 100). Clearly, \(14.24\) is greater than \(10\) but less than \(100\).
Thus, the estimated average rate of change of the function \(y = 16 \cdot 4^x\) is approximately \(14.24\) greater than the estimated average rate of change of the function \(y = 16x^2 + 4x\) over the interval \([1.1, 1.6]\).
Therefore, the result is: 10.