Chapter 4.1, Problem 110AYU

(a)

To determine

To draw: a scatter diagram of the given data.

Answer to Problem 110AYU

  

Explanation of Solution

Given:

Number of Cobalts Produced, x	Cost. C
0	10
1	23
2	31
3	38
4	43
5	50
6	59
7	70
8	85
9	105
10	135

Calculation:

Figure 1 shows the scatter diagram. A cubic relation may exist between the two variables x and C.

Figure 1

  

Conclusion:

Therefore, the scatter diagram is drawn.

(b)

To determine

To find: the cubic function of best fit

Answer to Problem 110AYU

The cubic equation of best fit to the data is $C\left(x\right)=0.2156x^3-2.347x^2+14.33x+10.22$

Explanation of Solution

Calculation:

Upon executing the CUBIC Regression Program, obtained results shown in Figure 2. The output that the utility provides shows us the equation $y=a{x}^3+b{x}^3+cx+d$ . The cubic equation of best fit to the data is

  $C\left(x\right)=0.2156x^3-2.347x^2+14.33x+10.22$

Figure 2

  

Conclusion:

Therefore, the cubic equation of best fit to the data is $C\left(x\right)=0.2156x^3-2.347x^2+14.33x+10.22$

(c)

To determine

To graph: the cubic function of best fit on the scatter diagram.

Answer to Problem 110AYU

  

Explanation of Solution

Calculation:

Figure 3 shows the graph of the cubic function of best fit on the scatter diagram. The function fits the data reasonably well. The cubic function of best fit passes through all the point very nearly.

Figure 3

  

Conclusion:

Therefore, the required graph is drawn.

(d)

To determine

To predict:the cost of manufacturing 11 Cobalt.

Answer to Problem 110AYU

The cost of manufacturing 11 cobalts is equal to 171.

Explanation of Solution

Calculation:

In Figure 1 point $x=11$ corresponds to 11 cobalt. The cost of manufacturing 11 cobalts can be calculated as

  $\begin{array}{c}C\left(x\right)=0.2156x^3-2.347x^2+14.33x+10.22\\ C\left(11\right)=0.2156\left({11}^3\right)-2.347\left({11}^2\right)+14.33\left(11\right)+10.22\\ =170.8266\\ \cong 171\end{array}$

Hence the cost of manufacturing 11 cobalts is equal to 171.

From Figure 3 at the point $x=11$ , C is approximately near to 171 and also the scatter diagram and cubic equation of best fit appear very nearby.

Conclusion:

Hence the cost of manufacturing 11 cobalts is equal to 171.

(e)

To determine

To interpret: the y-intercept.

Answer to Problem 110AYU

The y-intercept is $10.22$

Explanation of Solution

Calculation:

Find the y -intercept of the function

To find y -intercept

  $\begin{array}{c}C\left(x\right)=0.2156x^3-2.347x^2+14.33x+10.22\\ C\left(0\right)=0-0+0+10.22\\ =10.22\end{array}$

Conclusion:

Therefore, the y-intercept is $10.22$