To find the average price of ground beef during the first 3 months of the year, we first need to calculate the price at each month from [tex]\( t = 0 \)[/tex] to [tex]\( t = 3 \)[/tex] using the given price function:
[tex]\[
P(t) = 0.09 t^2 - 0.2 t + 4
\][/tex]
1. Calculate the price at [tex]\( t = 0 \)[/tex]:
[tex]\[
P(0) = 0.09(0)^2 - 0.2(0) + 4 = 4.0 \ \text{dollars per pound}
\][/tex]
2. Calculate the price at [tex]\( t = 1 \)[/tex]:
[tex]\[
P(1) = 0.09(1)^2 - 0.2(1) + 4 = 0.09 - 0.2 + 4 = 3.89 \ \text{dollars per pound}
\][/tex]
3. Calculate the price at [tex]\( t = 2 \)[/tex]:
[tex]\[
P(2) = 0.09(2)^2 - 0.2(2) + 4 = 0.09(4) - 0.2(2) + 4 = 0.36 - 0.4 + 4 = 3.96 \ \text{dollars per pound}
\][/tex]
4. Calculate the price at [tex]\( t = 3 \)[/tex]:
[tex]\[
P(3) = 0.09(3)^2 - 0.2(3) + 4 = 0.09(9) - 0.2(3) + 4 = 0.81 - 0.6 + 4 = 4.21 \ \text{dollars per pound}
\][/tex]
Now, to find the average price over the first 3 months (including [tex]\( t = 0 \)[/tex]), we compute the arithmetic mean of these prices:
[tex]\[
\text{Average price} = \frac{P(0) + P(1) + P(2) + P(3)}{4}
\][/tex]
Substituting the calculated values:
[tex]\[
\text{Average price} = \frac{4.0 + 3.89 + 3.96 + 4.21}{4}
\][/tex]
[tex]\[
\text{Average price} = \frac{16.06}{4} = 4.015 \ \text{dollars per pound}
\][/tex]
Therefore, the average price of ground beef during the first 3 months of the year is [tex]\( 4.015 \ \text{dollars per pound} \)[/tex].
