National Paper Company must purchase a new machine for producing cardboard boxes. The company must choose between two machines. The machines produce boxes of equal quality, so the company will choose the machine that produces (on average) the most boxes. It is known that there are substantial differences in the abilities of the company's machine operators. Therefore, National Paper has decided to compare the machines using a paired difference experiment. Suppose that eight randomly selected machine operators produce boxes for one hour using machine 1 and for one hour using machine 2, with the following results: Ds BoxYield

	Machine Operator
	1	2	3	4
Machine 1	53	60	58	48
Machine 2	50	55	56	44
	Machine Operator
	5	6	7	8
Machine 1	46	54	62	49
Machine 2	45	50	57	47

Assuming normality, perform a hypothesis test to determine whether there is a difference between the mean hourly outputs of the two machines. Use $\alpha=.05$

Round answer to the nearest tenth of a cent. Eduardo Diaz is a molding machine operator at US Accessories, Inc. He molds 130 tinted license plate holders from one container of molding plastic. Each contain er costs $5.85. Eduardo is able to mold 1 holder every 3 seconds. The direct labor cost is$18.95 per hour. What is the prime cost of manufacturing 1 tinted license plate holder?

E & T Excavation Company is planning an investment of $\$ 245,000$ for a bulldozer. The bulldozer is expected to operate for $1,500$ hours per year for five years. Customers will be charged $\$ 130$ per hour for bulldozer work. The bulldozer operator costs $\$ 32$ per hour in wages and benefits. The bulldozer is expected to require annual maintenance costing $\$ 15,000$. The bulldozer uses fuel that is expected to cost $\$ 42$ per hour of bulldozer operation.

Determine the equal annual net cash flows from operating the bulldozer.

An airplane appears on a radar tracking screen 55 mm to the right and 5 mm up from the lower left corner. At that same time, the operator notices a helicopter entering the screen 35 mm directly above the lower left corner of the screen. Each second the plane's position moves 2 mm left and 3 mm up and the helicopter's position moves 3 mm right and 1 mm up. Use this information. Write linear equations to represent their paths.

This exercise involves particular types of rotations in $\mathbb{R}^{3}$. (a) Explain why the mapping $L : \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by

$$L\left(\left[ \begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right]\right)=$$

$$\left[ \begin{array}{ccc}{\cos \theta} & {-\sin \theta} & {0} \\ {\sin \theta} & {\cos \theta} & {0} \\ {0} & {0} & {1}\end{array}\right] \left[ \begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right]$$

is a linear operator. (b) Show that the mapping L in part (a) rotates every vector in $\mathbb{R}^{3}$ about the z-axis through an angle of $\theta$ (as measured relative to the xy-plane). *(c) What matrix should be multiplied times [x,y,z] to create the linear operator that rotates $\mathbb{R}^{3}$ about the y-axis through an angle $\theta$ (relative to the xz-plane)? (Hint: When looking down from the positive y-axis toward the xz-plane in a right-handed system, the positive z-axis rotates $90^{\circ}$ counterclockwise into the positive x-axis.)

The Laplacian is the differential operator

$$\nabla^2=\nabla \cdot \nabla=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$

and Laplace's equation is

$$\nabla^2 w=\frac{\partial^2 w}{\partial x^2}+\frac{\partial^2 w}{\partial y^2}+\frac{\partial^2 w}{\partial z^2}=0 .$$

Any function that satisfies this equation is called harmonic. Show that the function $1 / f$ is harmonic.

Briggs Excavation Company is planning an investment of $132,000 for a bulldozer. The bulldozer is expected to operate for 1,500 hours per year for five years. Customers will be charged$110 per hour for bulldozer work. The bulldozer operator costs $28 per hour in wages and benefits. The bulldozer is expected to require annual maintenance costing$8,000. The bulldozer uses fuel that is expected to cost $46 per hour of bulldozer operation.

a. Determine the equal annual net cash flows from operating the bulldozer.

In an effort to determine the location of an operator site for a negatively regulated gene, you have made a series of deletions within the regulatory region. The extent of each deletion is shown by the line underneath the sequence, and the resulting expression from the operon (i = inducible; c = constitutive; - = no expression is also indicated).

. . . G G A T C T T A G C C G G C T A A C A T G A T A A A T A T A A . .

. . C C T A G A A T C G G C C G A T T G T A C T A T T T A T A T T . .

a. What can you conclude from these data about the location of the operator site?

b. Why do you think deletions 2 and 4 show no expression?

Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits, and let $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}$ be the distinct eigenvalues of T. For each $i,$ let $J_{i}$ be the Jordan canonical form of the restriction of $T$ to $\mathrm{K}_{\lambda_{i}} .$ Prove that $J=J_{1} \oplus J_{2} \oplus \cdots \oplus J_{k}$ is the Jordan canonical form of $J$

Let $B=([a, b],[c, d])$ be a basis for $\mathbb{R}^{2}$. Then $a d-b c \neq 0$ (why?). Let $L : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a linear operator such that $L([a, b])=[c, d]$ and $L([c, d])=[a, b]$. Show that the matrix for L with respect to the standard basis for $\mathbb{R}^{2}$ is

$$\frac{1}{a d-b c} \left[ \begin{array}{cc}{c d-a b} & {a^{2}-c^{2}} \\ {d^{2}-b^{2}} & {a b-c d}\end{array}\right]$$

.

Refer to the information presented in the preceding exercise.

Requirements

1. Prepare the journal entries to record the assignment of direct materials and direct labor and the allocation of manufacturing overhead to the Blending Department. Also, prepare the journal entry to record the costs of the gallons completed and transferred out to the Packaging Department. Assume labor costs are accrued and not yet paid.
2. Post the journal entries to the Work-in-Process Inventory-Blending T-account. What is the ending balance?
3. What is the average cost per gallon transferred out of the Blending Department into the Packaging Department? Why would the company managers want to know this cost?

Briggs Excavation Company is planning an investment of $132,000 for a bulldozer. The bulldozer is expected to operate for 1,500 hours per year for five years. Customers will be charged$110 per hour for bulldozer work. The bulldozer operator costs $28 per hour in wages and benefits. The bulldozer is expected to require annual maintenance costing$8,000. The bulldozer uses fuel that is expected to cost $46 per hour of bulldozer operation.

a. Calculate the equal annual net cash flows from operating the bulldozer.

Let $\mathrm{T}$ be a linear operator on a finite-dimensionial vector space $\mathrm{V}$ . and let $W_{1}, W_{2}, \ldots, W_{k}$ be T-invariant subspaces of $V$ such that $V=$ $W_{1} \oplus W_{2} \oplus \cdots \oplus W_{k}$ . Prove that $\operatorname{det}(T)=\operatorname{det}\left(T_{\mathrm{W}_{1}}\right) \operatorname{det}\left(T_{\mathrm{W}_{2}}\right) \cdots \operatorname{det}\left(T_{\mathrm{W}_{k}}\right)$

Let q be the quadratic form on $R^{2}$ given by $q\left(x_{1}, x_{2}\right)=a x_{1}^{2}+2 b x_{1} x_{2}+c x_{2}^{2}, a \neq 0$. Find an invertible linear operator U on $R^{2}$ such that $\left(U^{\dagger} q\right)\left(x_{1}, x_{2}\right)=a x_{1}^{2}+\left(c-\frac{b^{2}}{a}\right) x_{2}^{2}$