Question

the bar graph shows the estimated number of calories per day needed to maintain energy balance for various gender and age groups for sedentary lifestyles. (sedentary means a lifestyle that includes only the light physical activity associated with typical day - to - day life.) the mathematical model ( f=-81x^{2}+653x + 622 ) describes the number of calories needed per day ( f ) by females in age group ( x ) with sedentary lifestyles. according to the model, how many calories per day are needed by females between the ages of 9 and 13, inclusive, with this lifestyle? does this underestimate or overestimate the number shown by the graph? by how much?
according to the model, how many calories per day are needed by females between the ages of 9 and 13, inclusive, with this lifestyle?
1604 (type a whole number.)
does the model underestimate or overestimate the number shown by the graph? by how much?
the model ( f = - 81x^{2}+653x + 622 ) dropdown the actual amount by approximately blank calories
(type a whole number.)

Explanation:

Step1: Identify the age group x

For females between 9 - 13, from the graph's age groups, we can see that this corresponds to \( x = 2 \) (assuming the age groups are numbered, e.g., 4 - 8 is \( x = 1 \), 9 - 13 is \( x = 2 \), etc.).

Step2: Substitute x = 2 into the model

The model is \( F=-81x^{2}+653x + 622 \). Substitute \( x = 2 \):
\[

$$\begin{align*} F&=-81(2)^{2}+653(2)+622\\ &=-81\times4 + 1306+622\\ &=-324+1306 + 622\\ &=982+622\\ &=1604 \end{align*}$$

\]

Step3: Find the actual value from the graph

From the graph, for females in the 9 - 13 age group (Group 2), the number of calories is 1800? Wait, no, looking at the graph, the red bar (females) for 9 - 13 (Group 2) is 1800? Wait, the user's previous input had 1604 as the model's value. Wait, maybe the graph's value for females 9 - 13 is 1800? Wait, no, let's check again. Wait, the graph shows for age group 9 - 13 (Group 2), females (red) is 1800? Wait, the model gives 1604. So to find the difference: \( 1800 - 1604 = 196 \). Wait, but wait, maybe the graph's value is 1800? Wait, let's confirm. The model's value is 1604. If the graph's value for females 9 - 13 is 1800, then the model underestimates? Wait, no: model value is 1604, graph value is 1800. So \( 1800 - 1604 = 196 \). So the model underestimates by 196? Wait, no: if model is 1604 and actual is 1800, then model is less than actual, so it underestimates by \( 1800 - 1604 = 196 \).

Wait, maybe the graph's value for females 9 - 13 is 1800? Let's check the graph: the red bar (females) for 9 - 13 (Group 2) is 1800? The yellow bar (males) is 1700. So females: 1800. So model gives 1604. So difference is \( 1800 - 1604 = 196 \). So the model underestimates the actual amount by approximately 196 calories.

Answer:

The model \( F = -81x^{2}+653x + 622 \) underestimates the actual amount by approximately \(\boxed{196}\) calories.