Solve the quadratic equation to determine in which month Phoenix received the lowest amount of precipitation. The precipitation is modeled by

[tex]\[ y = -0.04088 x^2 + 0.4485 x + 1.862 \][/tex]

In what month did Phoenix receive the lowest amount of precipitation?

\begin{tabular}{|c|c|}
\hline
Month ( [tex]\( x \)[/tex] ) & Precipitation \\
\hline
January & 2.27 inches \\
\hline
February & ? \\
\hline
March & ? \\
\hline
April & ? \\
\hline
May & ? \\
\hline
June & ? \\
\hline
July & ? \\
\hline
August & ? \\
\hline
September & 2.59 inches \\
\hline
October & ? \\
\hline
November & ? \\
\hline
December & ? \\
\hline
\end{tabular}

To determine in which month Phoenix receives the lowest amount of precipitation, we need to find the vertex of the quadratic function [tex]\( y = -0.04088x^2 + 0.4485x + 1.862 \)[/tex].

A quadratic function in the form [tex]\( y = ax^2 + bx + c \)[/tex] has its vertex at [tex]\( x = -\frac{b}{2a} \)[/tex].

Given our function:

- [tex]\( a = -0.04088 \)[/tex]
- [tex]\( b = 0.4485 \)[/tex]
- [tex]\( c = 1.862 \)[/tex]

We substitute these values into the vertex formula:

[tex]\[ x = -\frac{b}{2a} \][/tex]

Substituting in the values, we get:

[tex]\[ x = -\frac{0.4485}{2 \times -0.04088} \][/tex]

This simplifies to:

[tex]\[ x = \frac{0.4485}{0.08176} \][/tex]

Performing the division gives:

[tex]\[ x \approx 5.49 \][/tex]

Since [tex]\( x \)[/tex] represents the month and we are considering the months as whole numbers, the closest month to 5.49 is the 6th month.

Thus, Phoenix receives the lowest amount of precipitation in the 6th month, which is June.

sqdancefan sqdancefan · Answer 2 · 2024-07-10T18:44:58-04:00

Answer:

December (1.36 inches)

Step-by-step explanation:

Given the precipitation model y = -0.04088x² +0.4485x +1.862, you want the know the month in which Phoenix received the least precipitation. A table is provided with January (2.27) and September (2.59) filled in.

Table

The attachment shows a spreadsheet with the rest of the precipitation table filled in according to the given formula. It is clear that December receives the least precipitation.

__

Additional comment

The leading coefficient of the quadratic is negative, so the equation describes a parabola that opens downward. At the value x = -b/(2a) ≈ 5.49, the precipitation will be maximum. Month 12 is farther from this maximum than month 1, so the precipitation will be lowest in month 12, December.