Chapter 6.4, Problem 72E

(a)

To determine

To find: the dot product $u·v$ and interpret the result in the context of the problem.

Answer to Problem 72E

The value is 11877.5.

Explanation of Solution

Given information:

The vector $u=\langle 3140,2750\rangle$ gives the numbers of hamburgers and hot dogs, respectively, sold at a fast food stand in one month. The vector $v=\langle 3140,2750\rangle$ gives the prices of the food items, respectively.

Calculation:

The dot product $u·v$ is given by,

  $u·v=\langle u_1,u_2\rangle ·\langle v_1,v_2\rangle$

Substituting the values we get,

  $\begin{array}{c}u·v=\langle 3140,2750\rangle ·\langle 2.25,1.75\rangle \\ =3140\left(2.25\right)+2750\left(1.75\right)\\ =7065+4812.5\\ =11877.5\end{array}$

Interpretation: The dot product $u·v$ gives us the total cost of food items sold by the fast food stand in one month.

(b)

To determine

To identify: the vector operation used to increase the prices by 2.5%.

Answer to Problem 72E

The new total cost of food items sold by the fast food stand in one month.

  $u·\left(1.025\right)v=\left(1.025\right)\left(u·v\right)$

Explanation of Solution

Calculation:

By rile of scalar multiplication,

  $\begin{array}{c}c\left(u·v\right)=cu·v\\ =u·cv\end{array}$

Therefore, to increase the prices by 2.5%, we use vector operation called, scalar multiplication.

Thus, gives the new total cost of food items sold by the fast food stand in one month.

  $u·\left(1.025\right)v=\left(1.025\right)\left(u·v\right)$

Hence,

1. The dot product $u·v$ gives us the total cost of food items sold by the fast food stand in one month.
2. To increase the prices by 2.5%, we use vector operation called, Scalar multiplication.