Question

1. the function ( g(x) = |0.05x - 0.85| + 1.77 ) models the gas price, in dollars, for january 6, 2020 to august 24, 2020, where ( x ) is the number of weeks since january 6, 2020. determine when gas prices were $2.52.

work
number of weeks since january 6, 2020

Explanation:

Step1: Set up the equation

We know that \( g(x) = |0.05x - 0.85| + 1.77 \) and \( g(x)=2.52 \). So we set up the equation:
\( |0.05x - 0.85| + 1.77 = 2.52 \)

Step2: Isolate the absolute value

Subtract \( 1.77 \) from both sides of the equation:
\( |0.05x - 0.85| = 2.52 - 1.77 \)
\( |0.05x - 0.85| = 0.75 \)

Step3: Solve the absolute value equation

The absolute value equation \( |A| = B \) (where \( B\geq0 \)) has two solutions: \( A = B \) or \( A=-B \). So we have two cases:

Case 1: \( 0.05x - 0.85 = 0.75 \)

Add \( 0.85 \) to both sides:
\( 0.05x = 0.75 + 0.85 \)
\( 0.05x = 1.6 \)
Divide both sides by \( 0.05 \):
\( x=\frac{1.6}{0.05}=32 \)

Case 2: \( 0.05x - 0.85=- 0.75 \)

Add \( 0.85 \) to both sides:
\( 0.05x=- 0.75 + 0.85 \)
\( 0.05x = 0.1 \)
Divide both sides by \( 0.05 \):
\( x=\frac{0.1}{0.05} = 2 \)

Now we need to check the time period. The function is for January 6, 2020 to August 24, 2020. Let's find the number of weeks between January 6, 2020 and August 24, 2020.

January has 31 days, from Jan 6 to Jan 31: \( 31 - 6=25 \) days.

February 2020 is a leap year? 2020 is divisible by 4, so February has 29 days.

March: 31, April:30, May:31, June:30, July:31, August 1 - 24:24 days.

Total days: \( 25+29 + 31+30+31+30+31+24=231 \) days. Number of weeks: \( \frac{231}{7}=33 \) weeks. So \( x = 32 \) is within the period (since 32 ≤ 33) and \( x = 2 \) is also within the period. But we need to see the context, probably we consider the non - trivial solution or the one that makes sense. But let's check the function. The function is a absolute value function, so we have two solutions. But let's verify:

For \( x = 2 \): \( g(2)=|0.05\times2 - 0.85|+1.77=|0.1 - 0.85|+1.77=| - 0.75|+1.77 = 0.75 + 1.77=2.52 \)

For \( x = 32 \): \( g(32)=|0.05\times32 - 0.85|+1.77=|1.6 - 0.85|+1.77=|0.75|+1.77 = 0.75+1.77 = 2.52 \)

Both are valid, but we need to see the time. January 6, 2020 plus 2 weeks is around January 20, 2020, and plus 32 weeks: 32 weeks is 224 days. January 6 + 224 days:

January: 31 - 6 = 25 days left in Jan.

February:29, March:31, April:30, May:31, June:30, July:31.

25+29+31+30+31+30+31 = 207 days. 224 - 207 = 17 days into August. So August 17, 2020, which is before August 24, 2020. So both are valid, but maybe the question expects both or one. But since the problem says "when", we can present both solutions. But let's check the calculation again.

Answer:

The number of weeks \( x \) is \( \boldsymbol{2} \) or \( \boldsymbol{32} \) (If we consider the context of the time period from January 6, 2020 to August 24, 2020, both \( x = 2 \) (around January 20, 2020) and \( x=32 \) (around August 17, 2020) are valid as they satisfy the equation \( g(x) = 2.52 \))