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Zorluk: OrtaLinear Equations in One Variable

Oven A preheats at a constant rate of 1515 degrees Celsius per minute, starting from an initial temperature of 2525 degrees Celsius. Oven B starts preheating 55 minutes after Oven A begins, starting from an initial temperature of 2020 degrees Celsius and preheating at a constant rate of 2020 degrees Celsius per minute. If both ovens continue to preheat, how many minutes after Oven A begins preheating will both ovens reach the same temperature?

Cevap: 21 minutes

Cevap

The ovens will reach the same temperature 2121 minutes after Oven A begins preheating.
To find the number of minutes after Oven A begins preheating when both ovens reach the same temperature, we can write an equation in terms of tt, the time in minutes since Oven A started. Oven A starts at 2525 degrees Celsius and increases by 1515 degrees per minute, so its temperature is 25+15t25 + 15t. Oven B starts 55 minutes later, meaning it preheats for t−5t - 5 minutes. Starting from 2020 degrees Celsius and preheating at 2020 degrees per minute, Oven B's temperature is 20+20(t−5)20 + 20(t - 5). Setting these two expressions equal gives the equation 25+15t=20+20(t−5)25 + 15t = 20 + 20(t - 5). Distributing 2020 yields 25+15t=20t−8025 + 15t = 20t - 80. Isolating tt gives 5t=1055t = 105, which results in t=21t = 21.

Adım Adım Çözüm

1
Set up expressions representing the temperature of each oven tt minutes after Oven A begins preheating.
Oven A: 25+15t25 + 15t; Oven B: 20+20(t−5)20 + 20(t - 5)
Since Oven B starts 55 minutes after Oven A, it has been preheating for t−5t - 5 minutes.
2
Set the temperature expressions equal to find the time at which they reach the same temperature.
25+15t=20+20(t−5)25 + 15t = 20 + 20(t - 5)
We want to find the value of tt where the temperatures of the two ovens are equal.
3
Distribute and simplify the equation.
25+15t=20t−8025 + 15t = 20t - 80
Distributing the 2020 across (t−5)(t - 5) yields 20t−10020t - 100. Combining the constant terms gives 20−100=−8020 - 100 = -80.
4
Solve for tt by isolating the variable term.
5t=1055t = 105, which gives t=21t = 21
Subtract 15t15t and add 8080 to both sides, then divide by 55.

Anahtar Kavram

Solving linear equations in one variable that model real-world situations with a time delay.
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Konular

Linear Equations in One Variable

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