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=-82x^{2}+653x + 621 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 4 and 8, 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 4 and 8, inclusive, with this lifestyle? (type a whole number.)

Explanation:

Step1: Identify the age - range and the formula

The age - range of females is \(4\leq x\leq8\), and the formula for the number of calories per day for sedentary females is \(F=- 82x^{2}+653x + 621\).

Step2: Substitute the values of \(x\) into the formula

We need to find the value of \(F\) when \(x\) is in the range \(4\leq x\leq8\). First, when \(x = 4\):
\[

$$\begin{align*} F&=-82\times4^{2}+653\times4 + 621\\ &=-82\times16+2612+621\\ &=-1312+2612+621\\ &=1300 + 621\\ &=1921 \end{align*}$$

\]
When \(x = 8\):
\[

$$\begin{align*} F&=-82\times8^{2}+653\times8+621\\ &=-82\times64 + 5224+621\\ &=-5248+5224+621\\ &=-24 + 621\\ &=597 \end{align*}$$

\]
We can also use the vertex formula for a quadratic function \(y = ax^{2}+bx + c\), the \(x\) - coordinate of the vertex is \(x=-\frac{b}{2a}\). Here \(a=-82\), \(b = 653\), so \(x=-\frac{653}{2\times(-82)}=\frac{653}{164}\approx3.98\). Since the coefficient of \(x^{2}\) (\(a=-82<0\)), the function is a downward - opening parabola. We can check the values of the function at the endpoints of the interval \([4,8]\).
We substitute \(x = 4\) into the formula \(F=-82x^{2}+653x + 621\):
\[

$$\begin{align*} F&=-82\times4^{2}+653\times4+621\\ &=-82\times16 + 2612+621\\ &=-1312+2612+621\\ &=1921 \end{align*}$$

\]

Answer:

1921