$\text { At time } t = 0 \text { minutes, the temperature of an object is } 160 ^ { \circ } \mathrm { F } \text {. The temperature of the }$ object is changing at the rate given by the differential equation $\frac { d y } { d t } = - \frac { 1 } { 2 } ( y - 92 )$. Use Euler's Method to approximate the particular solutions of this differential equation at $t = 2$. Use a step size of $h = 0.1$. Round your answer to one decimal place.
A) $123.5$
B) $120.4$
C) $121.9$
D) $128.7$
E) $116.4$

Question

Quizplus · Accepted Answer

Euler's Method involves using the formula $y_{n+1} = y_n + h \cdot f(t_n, y_n)$ to approximate solutions to differential equations. Given $\frac{dy}{dt} = -\frac{1}{2}(y - 92)$, an initial condition $y(0) = 160$, and a step size $h = 0.1$, we can calculate the temperature at $t = 2$ by iterating this process 20 times (since $2 / 0.1 = 20$). After performing these calculations, the approximation closest to the computed value is $116.4$, which corresponds to choice E.