Question

the function $g(x) = 1.7\sqrt{x} + 17.0$ models the median height, $g(x)$, in inches, of children who are $x$ months of age. the graph of $g$ is shown. graph with y-axis 0–40, x-axis 0–60, grid

the actual median height for children at 24 months is 25 inches. how well does the model describe the actual height?

the model describes the actual height
a. very well.
b. poorly.

c. use the model to find the average rate of change, in inches per month, between birth and 12 months.
the average rate of change is 0.5 inches per month. (round to the nearest tenth.)

d. use the model to find the average rate of change, in inches per month, between 40 and 52 months.
the average rate of change is 0.1 inches per month. (round to the nearest tenth.)

how does this compare with your answer in part (c)? how is this difference shown by the graph?
a. the average rate of change is larger. the graph is steeper.
b. the average rate of change is smaller. the graph is not as steep.
c. the average rate of change is larger. the graph is not as steep.
d. the average rate of change is smaller. the graph is steeper.

Explanation:

Part a (assuming the first part was about the model at 24 months, let's verify the model's prediction)

Step1: Substitute x = 24 into g(x)

We have the function \( g(x) = 1.7\sqrt{x}+17.0 \). Substitute \( x = 24 \):
\( g(24)=1.7\sqrt{24}+17.0 \)
First, calculate \( \sqrt{24}\approx4.9 \) (rounded to one decimal place).
Then, \( 1.7\times4.9 = 8.33 \).
So, \( g(24)\approx8.33 + 17.0 = 25.33 \) inches.

The actual height is 25 inches. The difference is \( |25.33 - 25| = 0.33 \) inches, which is small. So the model describes the actual height very well (Option A).

Part c: Average rate of change between birth (x=0) and 12 months (x=12)

Step1: Find g(0) and g(12)

For \( x = 0 \): \( g(0)=1.7\sqrt{0}+17.0 = 0 + 17.0 = 17.0 \) inches.
For \( x = 12 \): \( g(12)=1.7\sqrt{12}+17.0 \). \( \sqrt{12}\approx3.464 \), so \( 1.7\times3.464\approx5.89 \). Then \( g(12)\approx5.89 + 17.0 = 22.89 \) inches.

Step2: Calculate average rate of change

The average rate of change formula is \( \frac{g(b)-g(a)}{b - a} \), where \( a = 0 \), \( b = 12 \).
\( \frac{22.89 - 17.0}{12 - 0}=\frac{5.89}{12}\approx0.5 \) inches per month (matches the given answer).

Part d: Average rate of change between 40 and 52 months

Step1: Find g(40) and g(52)

For \( x = 40 \): \( g(40)=1.7\sqrt{40}+17.0 \). \( \sqrt{40}\approx6.325 \), so \( 1.7\times6.325\approx10.75 \). Then \( g(40)\approx10.75 + 17.0 = 27.75 \) inches.
For \( x = 52 \): \( g(52)=1.7\sqrt{52}+17.0 \). \( \sqrt{52}\approx7.211 \), so \( 1.7\times7.211\approx12.26 \). Then \( g(52)\approx12.26 + 17.0 = 29.26 \) inches.

Step2: Calculate average rate of change

Using the formula \( \frac{g(52)-g(40)}{52 - 40}=\frac{29.26 - 27.75}{12}=\frac{1.51}{12}\approx0.1 \) inches per month (matches the given answer).

Comparison (last part)

The average rate of change in part (c) is 0.5, and in part (d) is 0.1. So the rate in (d) is smaller. A smaller average rate of change means the graph is less steep (since slope = rate of change). So the correct option is B.

Final Answers:

• Part (a) Answer: A. very well.
• Part (c) Answer: 0.5 inches per month.
• Part (d) Answer: 0.1 inches per month.
• Comparison Answer: B. The average rate of change is smaller. The graph is not as steep.

Answer:

Step1: Find g(40) and g(52)

For \( x = 40 \): \( g(40)=1.7\sqrt{40}+17.0 \). \( \sqrt{40}\approx6.325 \), so \( 1.7\times6.325\approx10.75 \). Then \( g(40)\approx10.75 + 17.0 = 27.75 \) inches.
For \( x = 52 \): \( g(52)=1.7\sqrt{52}+17.0 \). \( \sqrt{52}\approx7.211 \), so \( 1.7\times7.211\approx12.26 \). Then \( g(52)\approx12.26 + 17.0 = 29.26 \) inches.

Step2: Calculate average rate of change

Using the formula \( \frac{g(52)-g(40)}{52 - 40}=\frac{29.26 - 27.75}{12}=\frac{1.51}{12}\approx0.1 \) inches per month (matches the given answer).

Comparison (last part)

The average rate of change in part (c) is 0.5, and in part (d) is 0.1. So the rate in (d) is smaller. A smaller average rate of change means the graph is less steep (since slope = rate of change). So the correct option is B.

Final Answers:

• Part (a) Answer: A. very well.
• Part (c) Answer: 0.5 inches per month.
• Part (d) Answer: 0.1 inches per month.
• Comparison Answer: B. The average rate of change is smaller. The graph is not as steep.