TMA manufactures 37-in. high-definition LCD televisions in two separate locations: Location I and Location II. The output at Location I is at most 6000 televisions/month, whereas the output at Location II is at most 5000 televisions/month. TMA is the main supplier of televisions to Pulsar Corporation, its holding company, which has priority in having all its requirements met. In a certain month, Pulsar placed orders for at most 3000 and at most 4000 televisions to be shipped to two of its factories located in City $A$ and City $B$, respectively. The shipping costs (in dollars) per television from the two TMA plants to the two Pulsar factories are as follows:

	To Pulsar Factories	To Pulsar Factories
From TMA	City $a$	City $b$
Location I	$\$ 6$	$\$ 4$
Location II	$\$ 8$	$\$ 10$

Find a shipping schedule that meets the requirements of both companies while keeping costs to a minimum.

In the following exercise, solve each linear programming problem by the method of corners.

$$\begin{array}{lr} \text { Maximize } & P=x+2 y \\ \text { subject to } & x+y \leq 4 \\ & 2 x+y \leq 5 \\ & x \geq 0, y \geq 0 \end{array}$$

Objective: Solve a linear programming application

a. Determine the values of x and y that produce the maximum or minimum value of the objective function on the given feasible region. b. Determine the maximum or minimum value of the objective function on the given feasible region. Maximize: z= 3x + 2y

City $A$ lies directly west of city $B$. When there is no wind, an airliner makes the 5550 -km round-trip flight between them in $6.60 \mathrm{~h}$ of flying time while traveling at the same speed in both directions. When a strong, steady $225-\mathrm{km} / \mathrm{h}$ wind is blowing from west to east and the airliner has the same airspeed as before, how long will the trip take?

In the following exercise, solve each linear programming problem by the method of corners.

$$\begin{array}{lr} \text { Minimize } & C=3 x+ y \\ \text { subject to } & x+2 y \geq 40 \\ & x+y \geq 30 \\ & x \geq 0, y \geq 0 \end{array}$$

Everest Deluxe World Travel has decided to advertise in the Sunday editions of two major newspapers in town. These advertisements are directed at three groups of potential customers. Each advertisement in Newspaper $\mathrm{I}$ is seen by 70,000 Group $A$ customers, 40,000 Group $B$ customers, and 20,000 Group $C$ customers. Each advertisement in Newspaper II is seen by 10,000 Group $A, 20,000$ Group $B$, and 40,000 Group $C$ customers. Each advertisement in Newspaper I costs $\$ 1000$, and each advertisement in Newspaper II costs $\$ 800$. Everest would like their advertisements to be read by at least 2 million people from Group $A, 1.4$ million people from Group $B$, and 1 million people from Group $C$. How many advertisements should Everest place in each newspaper to achieve its advertising goals at a minimum cost?

A pistol that fires a signal flare gives it an initial velocity (muzzle velocity) of $124~\frac{\text{m}}{\text{s}}$ at an angle of $50^{\circ}$ above the horizontal. You can ignore air resistance. Find the flare's maximum height.

A 1.00-kg object is observed to have an acceleration of $10.0 \mathrm{~m} / \mathrm{s}^2$ in a direction $30.0^{\circ}$ north of east. The force $\mathbf{F}_2$ acting on the object has a magnitude of $5.00 \mathrm{~N}$ and is directed north. Determine the magnitude and direction of the force $\mathbf{F}_1$ acting on the object.

A test rocket starting from rest at point A is launched by accelerating it along a 200.0-m incline at $1.90 \mathrm { m } / \mathrm { s } ^ { 2 }$. The incline rises at $35.0 ^ { \circ }$ above the horizontal, and at the instant the rocket leaves it, the engines turn off and the rocket is subject to gravity only (ignore air resistance). Find the rocket’s greatest horizontal range beyond point A.

A pistol that fires a signal flare gives it an initial velocity (muzzle velocity) of $125 \mathrm{~m} / \mathrm{s}$ at an angle of $55.0^{\circ}$ above the horizontal. You can ignore air resistance. Find the flare's maximum height and the distance from its firing point to its landing point if it is fired (a) on the level salt flats of Utah, and (b) over the flat Sea of Tranquility on the Moon, where $g=1.67 \mathrm{~m} / \mathrm{s}^2$.

A manufacturer of projection TVs must ship a total of at least 1000 TVs to its two central warehouses. Each warehouse can hold a maximum of $750 \mathrm{TVs}$. The first warehouse already has $150 \mathrm{TVs}$ on hand, whereas the second has $50 \mathrm{TVs}$ on hand. It costs $\$ 8$ to ship a TV to the first warehouse, and it costs $\$ 16$ to ship a TV to the second warehouse. How many TVs should be shipped to each warehouse to minimize the cost? What is the minimum cost?

A web page designer creates an animation in which a dot on a computer screen has position

$$\vec { r } = \left[ 4.0 \mathrm { cm } + \left( 2.5 \mathrm { cm } / \mathrm { s } ^ { 2 } \right) t ^ { 2 } \right] \hat { \imath } + ( 5.0 \mathrm { cm } / \mathrm { s } ) t \hat { \jmath }$$

. (a) Find the magnitude and direction of the dot’s average velocity between t = 0 and t = 2.0 s.(b) Find the magnitude and direction of the instantaneous velocity at t = 0, t = 1.0 s, and t = 2.0 s. (c) Sketch the dot’s trajectory from t = 0 to t = 2.0 s, and show the velocities calculated in part (b).

Refer to the previous exercise.

a. If the contribution to the profit of a model A grate is changed to $\$ 1.75 /$ grate, will the original optimal solution still hold? What will be the new optimal solution?

b. If the contribution to the profit of a model A grate is changed to $\$ 2.50$ /grate, will the original optimal solution still hold? What will be the new optimal solution?

Fill in the missing entries by performing the indicated row operations to obtain the row-reduced matrices.

$$\left[\begin{array}{rrr|r} 1 & 3 & 1 & 3 \\ 3 & 8 & 3 & 7 \\ 2 & -3 & 1 & -10 \end{array}\right] \underset{R_3-2 R_1}{\stackrel{R_2-3 R_1}{\longrightarrow}}\left[\begin{array}{lll|l} 1 & 3 & 1 & 3 \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \end{array}\right] \stackrel{-R_2}{\longrightarrow}$$

$$\left[\begin{array}{rrr|r} 1 & 3 & 1 & 3 \\ \cdot & \cdot & \cdot & \cdot \\ 0 & -9 & -1 & -16 \end{array}\right] \underset{R_3+9 R_2}{\stackrel{R_1-3 R_2}{\longrightarrow}}$$

$$\left[\begin{array}{ccc|c} \cdot & \cdot & \cdot & \cdot \\ 0 & 1 & 0 & 2 \\ \cdot & \cdot & \cdot & \cdot \end{array}\right] \underset{-R_3}{\stackrel{R_1+R_3}{\longrightarrow}}\left[\begin{array}{lll|r} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -2 \end{array}\right]$$