Question

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 = - 66x² + 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 the age - group value

Since we are considering women over the age of 51, we assume \(x = 51\).

Step2: Substitute \(x = 51\) into the formula

We have the formula \(W=-66x^{2}+526x + 1030\). Substitute \(x = 51\) into it:
First, calculate \(x^{2}=51^{2}=2601\).
Then, \(-66x^{2}=-66\times2601=- 171666\), \(526x=526\times51 = 26826\).
\[

$$\begin{align*} W&=-171666+26826 + 1030\\ &=-171666+(26826 + 1030)\\ &=-171666+27856\\ &=-143810 \end{align*}$$

\]
This is incorrect as we made a wrong start. Since we are just given \(W = 1810\) calories as the result from the model for \(x>51\) and we need to compare with the graph (but the graph - value is not given). Let's assume the graph - value is \(G\). If \(G>1810\), the model underestimates by \(G - 1810\) calories. If \(G<1810\), the model overestimates by \(1810 - G\) calories. Since the graph - value is not provided, we can't give a numerical answer for the difference. But we can say:
If the graph value is greater than 1810, the model underestimates. If the graph value is less than 1810, the model overestimates.

Answer:

If the value from the graph is \(G\):
If \(G>1810\), the model underestimates by \(G - 1810\) calories.
If \(G<1810\), the model overestimates by \(1810 - G\) calories.
Since the graph - value is not given, we can't fill in the blanks with a specific number.