To find how much time an 18-year-old spends watching television each day according to the given quadratic model, we need to substitute [tex]\( x = 18 \)[/tex] into the quadratic equation and solve for [tex]\( y \)[/tex].
The quadratic equation we are working with is:
[tex]\[
y = 0.004x^2 - 0.314x + 7.5
\][/tex]
Here, [tex]\( x \)[/tex] represents the age, and [tex]\( y \)[/tex] represents the number of hours spent watching television.
Let's substitute [tex]\( x = 18 \)[/tex] into the equation:
[tex]\[
y = 0.004(18)^2 - 0.314(18) + 7.5
\][/tex]
First, calculate [tex]\( 18^2 \)[/tex]:
[tex]\[
18^2 = 324
\][/tex]
Then multiply [tex]\( 0.004 \)[/tex] by [tex]\( 324 \)[/tex]:
[tex]\[
0.004 \times 324 = 1.296
\][/tex]
Next, multiply [tex]\( -0.314 \)[/tex] by [tex]\( 18 \)[/tex]:
[tex]\[
-0.314 \times 18 = -5.652
\][/tex]
Now, substitute these values back into the equation and simplify:
[tex]\[
y = 1.296 - 5.652 + 7.5
\][/tex]
Combine the terms:
[tex]\[
1.296 + 7.5 = 8.796
\][/tex]
[tex]\[
8.796 - 5.652 = 3.144
\][/tex]
Therefore, according to the quadratic model, an 18-year-old spends approximately [tex]\( 3.144 \)[/tex] hours watching television each day. Based on this result, the closest answer from the given choices is:
A. 3 hours
