Chapter 8.1, Problem 25E

a.

To determine

To calculate the total number of bagels sold over the 11-year period.

Answer to Problem 25E

The total number of bagels sold over 11-year period is 797.5 thousand.

Explanation of Solution

Given information:

year	Sales (thousands)
1995	5
1996	8.9
1997	16
1998	26.3
1999	39.8
2000	56.5
2001	76.4
2002	99.5
2003	125.8
2004	155.3
2005	188

Calculation:

Total no bagels sold

  $\begin{array}{l}=1000\left(5+8.9+16+26.3+39.8+56.5+76.5+99.5+125.8+155.3+188\right)\\ =797500\end{array}$

Therefore,

The total number of bagels sold over 11-year period is 797.5 thousand.

b.

To determine

To model annual bagel sales (in thousands) as a function of B(x) using quadratic regression, where x is the number of years after 1995.

Answer to Problem 25E

The model for annual bagel sales $B\left(x\right)=1.6x^2+2.3x+5$ where x is the number of years after 1995.

Explanation of Solution

Given information:

year	Sales (thousands)
1995	5
1996	8.9
1997	16
1998	26.3
1999	39.8
2000	56.5
2001	76.4
2002	99.5
2003	125.8
2004	155.3
2005	188

Calculation:

Assume the quadratic regression as

  $B\left(x\right)=a{x}^2+bx+c$

To evaluate a, band c put values of sales after 1 year of 1995

  $\begin{array}{l}8.9=a\cdot 1+b\cdot 1+c\\ 8.9=a+b+c\mathrm{...}\left(\text{i}\right)\\ \text{similarly for sales after 2 years of 1995}\\ 16=4a+2b+c\mathrm{...}\left(\text{ii}\right)\\ \text{and for sales after 3 years of 1995}\\ 26.3=9a+3b+c\mathrm{...}\left(\text{iii}\right)\\ \text{from equaton }\left(\text{i}\right)\\ 8.9-a-b=c\mathrm{...}\left(\text{iv}\right)\left[\text{ subtract}\left(a+b\right)\text{from each side}\right]\\ \text{substitute }\left(\text{iv}\right)\text{ in }\left(\text{ii}\right)\\ 16=4a+2b+8.9-a-b\\ 7.1=3a+b\mathrm{...}\left(\text{v}\right)\left[\text{simplify }\right]\\ \text{substitute }\left(\text{iv}\right)\text{ in }\left(\text{iii}\right)\\ 26.3=9a+3b+8.9-a-b\\ 17.4=8a+2b\\ b=\frac{17.4-8a}{2}\mathrm{...}\left(\text{vi}\right)\left[\text{simplify}\right]\\ \text{substitute }\left(\text{vi}\right)\text{ in }\left(\text{v}\right)\\ 7.1=3a+\frac{17.4-8a}{2}\\ 14.2=6a+17.4-8a\left[\text{multiply each side by 2}\right]\\ 2a=3.2\left[\text{simpilfy}\right]\\ a=1.6\left[\text{divide each side by 2}\right]\\ \text{now put the value of }a\text{ in equation }\left(\text{v}\right)\\ 7.1=3\cdot 1.6+b\\ 7.1=4.8+b\\ b=2.3\left[\text{subtract 4}\text{.8 from each side}\right]\\ \text{substitute the value of }a\text{ and }b\text{ in equation }\left(\text{iv}\right)\\ 8.9-1.6-2.3=c\\ c=5\end{array}$

Therefore,

The model for annual bagel sales $B\left(x\right)=1.6x^2+2.3x+5$ where x is the number of years after 1995.

c.

To determine

To integrate B(x)over the interval [0, 11] to find the bagel sales for 11 years.

Answer to Problem 25E

The bagel sales for 11 year is approximately 904.02 thousands

Explanation of Solution

Given information:

The interval of the integration is [0,11]

Calculation:

Integrate B(x)over the interval [0, 11]

  $\begin{array}{l}={\displaystyle \underset{0}{\overset{11}{\int }}(1.6x^2+2.3x+5})dx\\ ={\left[\frac{1.6}{3}{x}^3+\frac{2.3}{2}{x}^2+5x\right]}_0^{11}\\ ={\left[0.533x^3+1.15x^2+5x\right]}_0^{11}\\ \approx 904.02\end{array}$

Therefore,

The bagel sales for 11 year is approximately 904.02 thousands

d.

To determine

To explain graphically why the answer in part (a) is smaller than the answer in part (c).

Answer to Problem 25E

The answer in (a) corresponds to the area of left-hand rectangles. These rectangles lie under the curve B(x). The answer in (c) corresponds to the area under the curve this area is greater than the area of the rectangles.

Explanation of Solution

Given information:

Answer in part (a) is 797.5 thousands

Answer in part (c) is 904.02 thousands

The graph of the function is

  

The answer in (a) corresponds to the area of left-hand rectangles. These rectangles lie under the curve B(x). The answer in (c) corresponds to the area under the curve this area is greater than the area of the rectangles.