Question

1. At time \( t=0 \), a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is 91 degrees Celsius \( \left({ }^{\circ} \mathrm{C}\right) \) at time \( t=0 \), and the internal temperature of the potato is greater than \( 27^{\circ} \mathrm{C} \) for all times \( t>0 \). The internal temperature of the potato at time \( t \) minutes can be modeled by the function \( H \) that satisfies the differential equation \( \frac{d H}{d t}=-\frac{1}{4}(H-27) \), where \( H(t) \) is measured in degrees Celsius and \( H(0)=91 \). (a) Write an equation for the line tangent to the graph of \( H \) at \( t=0 \). Use this equation to approximate the internal temperature of the potato at time \( t=3 \). (b) Use \( \frac{d^{2} H}{d t^{2}} \) to determine whether your answer in part (a) is an underestimate or an overestimate of the internal temperature of the potato at time \( t=3 \). (c) For \( t<0 \), an alternate model for the internal temperature of the potato at time \( t \) minutes is the function \( G \) that satisfies the differential equation \( \frac{d G}{d t}=-(G-27)^{2 / 3} \), where \( G(t) \) is measured in degrees Celsius and \( G(0)=91 \). Find an expression for \( G(t) \). Based on this model, what is the internal temperature of the potato at time \( t=3 \) ?

          1. At time \( t=0 \), a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is 91 degrees Celsius \( \left({ }^{\circ} \mathrm{C}\right) \) at time \( t=0 \), and the internal temperature of the potato is greater than \( 27^{\circ} \mathrm{C} \) for all times \( t>0 \). The internal temperature of the potato at time \( t \) minutes can be modeled by the function \( H \) that satisfies the differential equation \( \frac{d H}{d t}=-\frac{1}{4}(H-27) \), where \( H(t) \) is measured in degrees Celsius and \( H(0)=91 \).
(a) Write an equation for the line tangent to the graph of \( H \) at \( t=0 \). Use this equation to approximate the internal temperature of the potato at time \( t=3 \).
(b) Use \( \frac{d^{2} H}{d t^{2}} \) to determine whether your answer in part (a) is an underestimate or an overestimate of the internal temperature of the potato at time \( t=3 \).
(c) For \( t<0 \), an alternate model for the internal temperature of the potato at time \( t \) minutes is the function \( G \) that satisfies the differential equation \( \frac{d G}{d t}=-(G-27)^{2 / 3} \), where \( G(t) \) is measured in degrees Celsius and \( G(0)=91 \). Find an expression for \( G(t) \). Based on this model, what is the internal temperature of the potato at time \( t=3 \) ?

1. At time t=0, a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is 91 degrees Celsius (^∘C) at time t=0, and the internal temperature of the potato is greater than 27^∘C for all times t>0. The internal temperature of the potato at time t minutes can be modeled by the function H that satisfies the differential equation (d H)/(d t)=-(1)/(4)(H-27), where H(t) is measured in degrees Celsius and H(0)=91.
(a) Write an equation for the line tangent to the graph of H at t=0. Use this equation to approximate the internal temperature of the potato at time t=3.
(b) Use (d^2 H)/(d t^2) to determine whether your answer in part (a) is an underestimate or an overestimate of the internal temperature of the potato at time t=3.
(c) For t<0, an alternate model for the internal temperature of the potato at time t minutes is the function G that satisfies the differential equation (d G)/(d t)=-(G-27)^2 / 3, where G(t) is measured in degrees Celsius and G(0)=91. Find an expression for G(t). Based on this model, what is the internal temperature of the potato at time t=3 ?

Added by Mar L.

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