Let's break down the problem step by step to determine which of the given inequalities accurately describes the situation.
The problem states that after 7 a.m., the initial power usage is 15,040 kWh. The goal is to determine the number of hours, [tex]\( t \)[/tex], such that the power usage does not exceed 100,000 kWh.
Each inequality represents a different exponential growth factor. Our objective is to identify the appropriate growth rate that models the power usage over time.
Let's consider each option provided:
- Option A: [tex]\(15,040(1.021)^t \leq 100,000\)[/tex]
- Option B: [tex]\(15,040(1.79)^t \leq 100,000\)[/tex]
- Option C: [tex]\(15,040(0.79)^t \leq 100,000\)[/tex]
- Option D: [tex]\(15,040(1.21)^t \leq 100,000\)[/tex]
To identify the correct inequality, we need to consider the nature of power usage growth, which typically does not decrease (ruling out any values effectively below 1, like 0.79) and often involves a growth factor that reasonably escalates the initial value toward the threshold (100,000 kWh).
Given the need to compare these scenarios and understanding typical problem structures,
- Option A: uses a growth factor of 1.021.
- Option B: massively multiplies 15,040, growing quickly.
- Option C: actually shows a decay due to the factor being less than 1 (0.79).
- Option D: represents more modest growth than option B but faster than option A.
Based on this, the ideal inequality balances steady exponential growth without violating the upper limit too quickly.
Therefore, Option A, [tex]\(15,040(1.021)^t \leq 100,000\)[/tex] is the correct choice. It accurately models the needed balance between initial value, gradual growth factor, and the limiting threshold.
Conclusively, the correct inequality is:
[tex]\[ 15,040(1.021)^t \leq 100,000 \][/tex]
Thus, the correct answer is:
Option A: [tex]\(15,040(1.021)^t \leq 100,000 \)[/tex]
