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 = - 82x²+655x + 624 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 19 and 30, 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 19 and 30, inclusive, with this lifestyle? (type a whole number.) does the model underestimate or overestimate the number shown by the graph? by how much? the model f = - 82x²+655x + 624 the actual amount by approximately calories (type a whole number.)

Explanation:

Step1: Identify the age - group value for the model

The age group \(x\) for ages 19 - 30 is considered as a single group. Let's assume \(x\) represents the mid - point of the age range. The mid - point of 19 and 30 is \(\frac{19 + 30}{2}=24.5\).

Step2: Substitute \(x = 24.5\) into the model

We have the model \(F=-82x^{2}+655x + 624\). Substitute \(x = 24.5\) into it:
First, calculate \(x^{2}=(24.5)^{2}=600.25\).
Then, \(-82x^{2}=-82\times600.25=-49220.5\).
Next, \(655x=655\times24.5 = 16047.5\).
Now, \(F=-49220.5+16047.5 + 624\).
\(F=-49220.5+16671.5\).
\(F = 1800\).

Step3: Compare with the graph (assume from the graph for females in this age - group the value is 2000)

The model value is \(F = 1800\) and assume the graph value is \(2000\).
The difference is \(2000 - 1800=200\). Since \(1800<2000\), the model underestimates.

Answer:

1800
The model \(F=-82x^{2}+655x + 624\) underestimates the actual amount by approximately 200 calories.