A cup of coffee is cooling down in a room following the equation x = 15 + 70e^(-t/40). Find the rate at which the temperature is decreasing when the coffee cools to 60°C.

First, we need to find the value of t when x = 6p°C. We are told that after t minutes the temperature, x, will be 60°C; so we can insert 60 into the equation for x: 

60 = 15 + 70e^(-t/40)

Secondly, we can rearrange the equation to get like terms on each side, meaning we subtract 15 from both sides. 

60 - 15 = 15 + 70e^(-t/40) - 15

45 = 70e^(-t/40)

Thirdly, we can divide both sides by 70 to get the 'e' term on its own. This will make the final step for this part of the question easier, but isn't necessarily needed at this stage:

45/70 = (70e^(-t/40))/70

Simplify: 9/14 = e^(-t/40)

Fourthly, take the ln of both sides to remove the e function, and divide by -1/40 to isolate t:

ln(9/14) = ln(e^(-t/40))

ln(9/14) = -t/40

-40ln(9/14)  = t

t =~ 17.67 mins

To solve the second part of the question we first need to differentiate the initial equation:

x = 15 + 70e^(-t/40)

The differential of an exponential function is the first derivative of the term the function is applied to, -t/40. Differentiating this with respect to t is simply -1/40. Remembering the product rule tells us to multiply this by the initial 70:

dx/dt = -70/40e^(-t/40)

Simplify: dx/dt = -7/4e^(-t/40)

Finally, substitute the saved value of t into this equation:

dx/dt = -7/4e^(-(-40ln(9/14))/40)

        = -9/8°C/min

Therefore the temperature is decreasing at 9/8°C/min. Remember the question asks for the rate of decrease so the answer should be positive. You may lose marks if you leave the answer negative. 

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