Question

question
nathan is deciding between two landscaping companies for his place of business. company a charges $75 per hour and a $150 equipment fee. company b charges $50 per hour and a $350 equipment fee. let a represent the amount company a would charge for t hours of landscaping, and let b represent the amount company b would charge for t hours of landscaping. graph each function and determine the interval of hours, t, for which company a is cheaper than company b.

click twice to plot each line. click a line to delete it.

answer attempt 1 out of 2
company a is cheaper than company b when t
note: the interval will only apply for valid values of t in the domain.
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Explanation:

Step1: Define the cost functions

Company A's cost: \( A = 75t + 150 \) (75 per hour, 150 equipment fee)
Company B's cost: \( B = 50t + 350 \) (50 per hour, 350 equipment fee)

Step2: Find when \( A < B \)

Set up the inequality: \( 75t + 150 < 50t + 350 \)
Subtract \( 50t \) from both sides: \( 25t + 150 < 350 \)
Subtract 150 from both sides: \( 25t < 200 \)
Divide by 25: \( t < 8 \)
Since \( t \) represents hours (valid domain \( t \geq 0 \)), the interval is \( 0 \leq t < 8 \) (or \( t < 8 \) with \( t \geq 0 \) implied).

Answer:

Company A is cheaper than Company B when \( t < 8 \) hours.