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Difficulty: MediumLinear Functions and Graphs

A factory manufactures solar panels at a constant rate. At the start of a morning shift, the factory has already manufactured 8080 solar panels. The total number of solar panels manufactured tt hours after the shift begins is modeled by a linear function. If the factory has manufactured a total of 200200 solar panels 33 hours after the shift begins, how many hours after the shift begins will the factory have manufactured a total of 440440 solar panels?

Answer: 9 hours

Answer

9
To find the time when the total number of solar panels reaches 440440, we first determine the constant rate of production. The change in solar panels over the first 33 hours is 200−80=120200 - 80 = 120 panels. Dividing by the 33 hours gives a constant rate of 4040 panels per hour. The linear equation representing the total panels is N(t)=40t+80N(t) = 40t + 80. Setting N(t)=440N(t) = 440 gives 440=40t+80440 = 40t + 80, which simplifies to 360=40t360 = 40t. Solving for tt yields t=9t = 9.

Step-by-Step Solution

1
Calculate the constant rate of production (slope) of the linear function.
The rate is 4040 solar panels per hour.
The factory starts with 8080 solar panels and reaches 200200 solar panels in 33 hours. The rate of change is the change in the number of panels divided by the change in time: 200−803−0=1203=40\frac{200 - 80}{3 - 0} = \frac{120}{3} = 40 panels per hour.
2
Write the linear equation representing the total number of solar panels manufactured, N(t)N(t), after tt hours.
N(t)=40t+80N(t) = 40t + 80
Since the initial quantity is 8080 and the rate of production is 4040 panels per hour, the linear function is N(t)=40t+80N(t) = 40t + 80.
3
Solve for the time tt when the total number of solar panels is 440440.
t=9t = 9 hours
Set N(t)=440N(t) = 440 in the equation: 440=40t+80440 = 40t + 80. Subtract 8080 from both sides to get 360=40t360 = 40t. Divide both sides by 4040 to find t=9t = 9.

Key Concept

Linear Functions and Graphs
Estimated Time:1m 30s
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Topics

Linear Functions and Graphs

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