Video: Estimating Using an Exponential Regression Model in Context

Michael made a pot of tea and then recorded its temperature, in degrees Fahrenheit, over the following hour. Michael concluded that the temperature 𝑇 of the tea, 𝑥 minutes after it is made, can be modelled by the exponential regression model 𝑇 = 190 × 0.985^𝑥. Use the model to estimate, to the nearest degree, the temperature of the tea after 25 minutes.

02:35

Video Transcript

Michael made a pot of tea and then recorded its temperature, in degrees Fahrenheit, over the following hour. Michael concluded that the temperature 𝑇 of the tea, 𝑥 minutes after it is made, can be modelled by the exponential regression model 𝑇 equals 190 multiplied by 0.985 to the power of 𝑥. Use the model to estimate to the nearest degree the temperature of the tea after 25 minutes.

So the question actually states that Michael is gonna be using a regression model, so an exponential regression model, to actually predict the values of the temperature of his tea at certain times. The model that he’s using is 𝑇 is equal to 190 multiplied by 0.985 to the power of 𝑥. And what we need to do is actually use this model to predict the temperature after 25 minutes.

Well, to do that, we then know that our 𝑥-value is actually going to be 25. And that’s because we’re told that we want to find the temperature of the tea after 25 minutes. And we’re told then in the question that 𝑥 is the minutes after it is made. So what we’re gonna do now is actually substitute in our 𝑥 is equal to 25 into our regression model. And when we do that, we get 𝑇 is equal to 190 multiplied by 0.985 to the power of 25.

And when we actually put this into our calculator, we’re gonna get this equal to 130.2145047. And the way we actually put that into our calculator was using this button here: you got an 𝑥 then with a little square in the exponential area. And that’s how we’d actually use it because we’d actually type 190 then multiply it by 0.985 then press this button and then press 25. Now, it’ll give you the calculation you want.

Well, that’s great. Have we finished here? Well, no, there’s one more step. And that’s because if we look back to the question, it says we want it left to the nearest degree.

So therefore, we can say that if Michael made a pot of tea and then recorded its temperature in degrees Fahrenheit over the following hour and then he concluded that the temperature 𝑇 of the tea 𝑥 minutes after it is made can be modelled by the exponential regression Model 𝑇 equals 190 multiplied by 0.985 to the power of 𝑇, then after 25 minutes his tea is gonna be 130 degrees and that’s to the nearest degree.

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