To find the maximum point of the quadratic equation [tex]\(P(t) = -16t^2 + 2050t + 150,\)[/tex] we follow these steps:
1. Identify the coefficients: The given quadratic equation is in the form [tex]\(P(t) = at^2 + bt + c\)[/tex], where:
- [tex]\(a = -16\)[/tex]
- [tex]\(b = 2050\)[/tex]
- [tex]\(c = 150\)[/tex]
2. Find the vertex of the parabola: The maximum or minimum value of a parabola [tex]\(ax^2 + bx + c\)[/tex] occurs at its vertex. For a quadratic function, the time at which the maximum profit occurs can be found using the formula:
[tex]\[
t = -\frac{b}{2a}
\][/tex]
3. Substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula:
[tex]\[
t = -\frac{2050}{2 \times -16}
\][/tex]
4. Calculate the value of [tex]\(t\)[/tex]:
[tex]\[
t = -\frac{2050}{-32}
\][/tex]
[tex]\[
t = 64.0625
\][/tex]
5. Determine the maximum profit by substituting [tex]\(t\)[/tex] back into the original equation:
[tex]\[
P(64.0625) = -16(64.0625)^2 + 2050(64.0625) + 150
\][/tex]
6. Calculate each term:
- First term:
[tex]\[
-16 \times (64.0625)^2 = -16 \times 4104.00390625 = -65664.0625
\][/tex]
- Second term:
[tex]\[
2050 \times 64.0625 = 131728.125
\][/tex]
- Third term:
[tex]\[
150
\][/tex]
7. Add the results to find the maximum profit:
[tex]\[
P(64.0625) = -65664.0625 + 131728.125 + 150 = 65814.0625
\][/tex]
Therefore, the maximum profit [tex]\(P(t)\)[/tex] is $65814.0625, and it occurs at [tex]\(t = 64.0625\)[/tex] days.
In conclusion, the beekeeper should wait approximately 64.0625 days to harvest the honey in order to maximize profit.
Therefore, the correct answer is that the beekeeper should wait 64.0625 days to maximize her profit, and the maximum profit is 65814.0625.
