the temperature, T (celcius) of a vehicle engine after being turned off for t minutes is modeled as: T(t)=100e^-kt -15 determine the following if the temperature is 45 degrees celcius after 10minutes. cooling rate (k)= initial temperature (T) at t=0 time (t) when temperature T=0 steady state temperature (T) after a long time.
Added by Paul H.
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We know that \( T(10) = 45 \) degrees Celsius. We can substitute \( t = 10 \) into the temperature model: \[ T(10) = 100e^{-10k} - 15 \] Setting this equal to 45: \[ 100e^{-10k} - 15 = 45 \] Adding 15 to both sides: \[ 100e^{-10k} = 60 \] Dividing both Show more…
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Frank's automobile engine runs at $100^{\circ} \mathrm{C}$. On a day when the outside temperature is $21^{\circ} \mathrm{C}$, he tums off the ignition and notes that 5 minutes later, the engine has cooled to $70^{\circ} \mathrm{C}$. (a) Determine the engine's cooling constant $k$. (b) What is the formula for $y(t) ?$ (c) When will the engine cool to $40^{\circ} \mathrm{C} ?$
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Question 1. (4 marks) A car engine running at a temperature of 80°C is then switched off and begins to cool according to Newton's Law of Cooling. The ambient temperature is 15°C, and after exactly 18 minutes and 9 seconds the engine temperature has reduced to 50°C. (a) Write down a model for the cooling of the engine using Newton's Law of Cooling in the form T(t) = Ts + (T0 - Ts)e^-kt where t is time measured in minutes, T0 is the initial temperature of the engine (in degrees Celsius), Ts is the ambient temperature (in degrees Celsius), T(t) is the engine temperature (in degrees Celsius) at time t and k is a rate constant. Here constructing the model involves substituting the initial engine temperature and the ambient temperature into the given equation for Newton's Law of Cooling, and then simplifying the equation where possible. (b) Hence determine the rate constant, k. (c) How long after the engine was turned off does the engine temperature drop to 30°C?
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13. Frank's automobile engine runs at 100°C. On a day when the outside temperature is 21°C, he turns off the ignition and notes that 5 minutes later, the engine has cooled to 70°C. Determine the engine's cooling constant k. What is the formula for y(t)? When will the engine cool to 40°C?
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