Problem

Suppose a metal block is cooling so that its temperature $T$ (in ${ }^{\circ} \mathrm{C}$ ) is given by $T=200 \cdot 4^{-0.1 t}+20$, where $t$ is in hours. a. Find the temperature after (i) 2 hours, (ii) 3.5 hours. b. How long has the cooling been taking place if the block now has a temperature of $120^{\circ} \mathrm{C}$ ? c. Find the eventual temperature $(t \rightarrow \infty)$. a. (i) After 2 hours the temperature will be about $171.6^{\circ} \mathrm{C}$. (Simplify your answer. Do not round until the final answer. Then round to the nearest tenth as needed.) (ii) After 3.5 hours the temperature will be about $\square^{\circ} \mathrm{C}$. (Simplify your answer. Do not round until the final answer. Then round to the nearest tenth as needed.)

Solution

Step 1 :To find the temperature after 3.5 hours, we need to substitute \(t=3.5\) into the given equation \(T=200 \cdot 4^{-0.1 t}+20\).

Step 2 :Substituting \(t=3.5\) into the equation, we get \(T = 200 \cdot 4^{-0.1 \cdot 3.5}+20\).

Step 3 :Solving the equation, we get \(T = 143.11444133449163\).

Step 4 :Rounding to the nearest tenth, we get \(T = 143.1\).

Step 5 :Final Answer: After 3.5 hours the temperature will be about \(\boxed{143.1^{\circ} \mathrm{C}}\).

From Solvely APP
Source: https://solvelyapp.com/problems/20448/

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