Chapter 4.1, Problem 111AYU

(a)

To determine

To draw: a scatter diagram of the given data.

Answer to Problem 111AYU

  

Explanation of Solution

Given:

Hours after Midnight, x	Temperature (°F), T
3	45.0
6	44.1
9	51.1
12	57.9
15	63.0
18	63.0
21	59.0
24	54.0

Calculation:

The scatter diagram for the above data is as follows.

  

By observing the above graph it can be seen that the relation between the variables is cubic.

Conclusion:

Therefore, the scatter diagram is drawn.

(b)

To determine

To find: the average rate of change in temperature

Answer to Problem 111AYU

The average rate of change in temperature is $2.27°\text{F/h}$

Explanation of Solution

Calculation:

The change in temperature from 9 AM to 12 noon is as follows

  $57.9°\text{F}-51.1°\text{F}=6.8°\text{F}$

Therefore the average rate of change in temperature is $6.8°\text{F}÷3\text{h}=2.27°\text{F/h}$

Conclusion:

Therefore the average rate of change in temperature is $2.27°\text{F/h}$

(c)

To determine

To find: the average rate of change in temperature

Answer to Problem 111AYU

Therefore the average rate of change in temperature is $0°\text{F/h}$

Explanation of Solution

Calculation:

The change in temperature from 3 PM to 6 PM is as follows

  $63.0°\text{F}-63.0°\text{F}=0°\text{F}$

Therefore the average rate of change in temperature is $0°\text{F/h}$

Conclusion:

Therefore the average rate of change in temperature is $0°\text{F/h}$

(d)

To determine

To decide: a function of best fit and predict the temperature at 5 PM.

Answer to Problem 111AYU

The temperature at 5 PM is $63.044°\text{F}$ .

Explanation of Solution

Calculation:

By observing the scatter diagram of the data it can be seen that the cubic function will model the relation between the variables. The cubic function obtained by the graphic utility is as follows

  $c\left(x\right)=45.393-1.374x+0.317x^2-0.0103x^3$

To predict the temperature at 5 PM. substitute the value $x=17$ in $c\left(x\right)$

  $\begin{array}{c}c\left(17\right)=45.393-1.374\times 17+0.317\times {17}^2-0.0103\times {17}^3\\ =45.393-23.358+91.613-50.6039\\ =63.044\end{array}$

Therefore, the temperature at 5 PM is $63.044°\text{F}$ .

Conclusion:

Therefore, the temperature at 5 PM is $63.044°\text{F}$ .

(e)

To determine

To draw: a scatter diagram of the given data.

Answer to Problem 111AYU

  

Explanation of Solution

Calculation:

The graph of cubic function obtained by graphic utility is as follows.

  

Conclusion:

Therefore, the scatter diagram is drawn.

(f)

To determine

To interpret: the y-intercept.

Answer to Problem 111AYU

The y-intercept is $45.393$ .

Explanation of Solution

Calculation:

By observing the graph and from the cubic equation, it is obtained that the y- intercept is $45.393$ .

Conclusion:

Therefore, the y-intercept is $45.393$ .