Question

a can of soda is placed inside a cooler. as the soda cools, its temperature t(x) in degrees celsius is given by the following function, where x is the number of minutes since the can was placed in the cooler. t(x)= - 17 + 39e^(-0.034x). find the initial temperature of the soda and its temperature after 20 minutes. round your answers to the nearest degree as necessary. initial temperature: □ °c temperature after 20 minutes: □ °c

Explanation:

Step1: Find initial temperature

When \(x = 0\), substitute into \(T(x)=- 17+39e^{-0.034x}\). Since \(e^{0}=1\), we have \(T(0)=-17 + 39\times1\).
\[T(0)=-17 + 39=22\]

Step2: Find temperature after 20 minutes

Substitute \(x = 20\) into \(T(x)=-17 + 39e^{-0.034x}\). First, calculate the exponent part: \(-0.034\times20=-0.68\). Then find \(e^{-0.68}\approx0.5066\). Now, \(T(20)=-17+39\times0.5066\). \(39\times0.5066 = 19.7574\), and \(T(20)=-17 + 19.7574=2.7574\approx3\).

Answer:

Initial temperature: \(22^{\circ}C\)
Temperature after 20 minutes: \(3^{\circ}C\)