Chapter 8.2, Problem 30PPS

(a)

To determine

To find: the temperature of the hot chocolate after $15\min$.

Answer to Problem 30PPS

The temperature of the hot chocolate after $15\min$ is $44.99°$.

Explanation of Solution

Given information:

The temperature model of hot chocolate is given by $y\left(t\right)=20+70{\left(1.071\right)}^{-t}$.

Consider the equation.

  $y\left(t\right)=20+70{\left(1.071\right)}^{-t}$.

Substitute $15$ for $t$ in the temperature model $y\left(t\right)=20+70{\left(1.071\right)}^{-t}$ to find the value of $y$.

  $\begin{array}{c}y\left(15\right)=20+70{\left(1.071\right)}^{-15}\\ =20+70\left(0.357\right)\\ =20+24.990\\ =44.990\end{array}$

Therefore, the temperature of the hot chocolate after $15\min$ is $44.99°$.

(b)

To determine

To find: the temperature of the hot chocolate after $30\min$.

Answer to Problem 30PPS

The temperature of the hot chocolate after $30\min$ is $28.960°$.

Explanation of Solution

Given information:

The temperature model of hot chocolate is given by $y\left(t\right)=20+70{\left(1.071\right)}^{-t}$.

Consider the equation.

  $y\left(t\right)=20+70{\left(1.071\right)}^{-t}$.

Substitute $30$ for $t$ in the temperature model $y\left(t\right)=20+70{\left(1.071\right)}^{-t}$ to find the value of $y$.

  $\begin{array}{c}y\left(30\right)=20+70{\left(1.071\right)}^{-30}\\ =20+70\left(0.128\right)\\ =20+8.960\\ =28.960\end{array}$

Therefore, the temperature of the hot chocolate after $30\min$ is $28.960°$.

(c)

To determine

To find: the temperature of the hot chocolate after $10\min$.

Answer to Problem 30PPS

The yes, the temperature of mug will be below the optimum drinking temperature after $10$ minutes.

Explanation of Solution

Given information:

The temperature model of hot chocolate is given by $y\left(t\right)=20+70{\left(1.071\right)}^{-t}$.

Consider the equation.

  $y\left(t\right)=20+70{\left(1.071\right)}^{-t}$.

Substitute $10$ for $t$ in the temperature model $y\left(t\right)=20+70{\left(1.071\right)}^{-t}$ to find the value of $y$.

  $\begin{array}{c}y\left(10\right)=20+70{\left(1.071\right)}^{-10}\\ =20+70\left(0.504\right)\\ =20+35.280\\ =55.280\end{array}$

As $55.280<60$ . So, the mug will be below the optimum drinking temperature after $10$ minutes.

Therefore, yes, the temperature of mug will be below the optimum drinking temperature after $10$ minutes.