Question

the bar graph shows the estimated number of calories per day needed to maintain energy balance for various gender and age groups for moderately active lifestyles. moderately active means a lifestyle that includes physical activity equivalent to walking 1.5 to 3 miles per day at 3 to 4 miles per hour, in addition to the light physical activity associated with typical day - to - day life. complete parts a. through c. a. the mathematical model below describes the number of calories needed per day, w, by women in age group x with moderately active lifestyles. w=-66x^{2}+526x + 1030. according to the model, how many calories per day are needed by women over the age of 51, with this lifestyle? w = 1810 calories. does this underestimate or overestimate the number shown by the graph? by how much? the model the number shown by the graph by calorie(s).

Explanation:

Step1: Identify age - group value

For women over 51, we assume \(x = 6\) (since usually age - groups are numbered, and if we consider the last group as \(x = 6\)).

Step2: Substitute \(x\) into the formula

Substitute \(x = 6\) into \(W=-66x^{2}+526x + 1030\).
\[

$$\begin{align*} W&=-66\times6^{2}+526\times6 + 1030\\ &=-66\times36+3156+1030\\ &=-2376+3156 + 1030\\ &=780+1030\\ &=1810 \end{align*}$$

\]

Step3: Read value from the graph

From the graph, for women in the age - group 51+, the number of calories is 1600.

Step4: Compare model and graph values

Calculate the difference \(1810−1600 = 210\). Since \(1810>1600\), the model over - estimates.

Answer:

The model over - estimates the number shown by the graph by 210 calories.