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Problem 70

Time In Exercise 39 of Section $9.3,$ we saw that the time (in hours) that a branch of Amalgamated Entities needs to spend to meet the quota set by the main office can be approximated by

$T(x, y)=x^{4}+16 y^{4}-32 x y+40$

where $x$ represents how many thousands of dollars the factory pends on quality control and $y$ represents how many thousands of dollars they spend on consulting. Find the average time if the

mount spent on quality control varies from $\$ 0$ to $\$ 4000$ andthe amount spent on consulting varies from $\$ 0$ to $\$ 2000 .$ Hint: Refer to Exercises $61-64$ .

Answer

78.4 $\mathrm{hr}$

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## Discussion

## Video Transcript

Okay, so here we have the function t X lie. It's next to the fourth sixteen White of the fourth, minus thirty two x. Why plus forty and t is time and X is thousands of dollars on quality control wise, thousands of dollars and consulting. So we're looking over this rectangle, actually, is your four for Pax and zero to why we want to know the outers values to the area of this rectangle is eight. So we're going to divide by the area that rectangle one eighth and then we just want toe integrate the function over this region. So taking anti derivative with respect to why so that'LL be why X to fourth plus sixteen fifth slide minus sixteen X squared plus forty. Why? Evaluated from zero to two. Okay, this is one eighth you're in for a worser is going going into two x to the fourth. Okay, so this is sixteen to to the fifth is thirty two over five minus. Uh, see, it's why I squared. So sixteen times for sixty four X plus eighty the ex. Okay, so now I'll take another antiterror evident. Uh, okay. We have two fifths x to the fifth. And then whatever this, well, we can think about this. So this is to force to death. Certitude of the ninth. It's five twelve. Okay, there, minus thirty two X squared eighty X value, mate from zero to four. So we really just need a plug in for you. Look so right. And what do we get? We get seventy eight point four hours is the average time, Which is you could tio seventy hours in twenty four, Dennis.

## Recommended Questions

Time In an exercise earlier in this chapter, we saw that the time (in hours) that a branch of Amalgamated Entities needs to spend to meet the quota set by the main office can be approximated by

$$T(x, y)=x^{4}+16 y^{4}-32 x y+40$$

where $x$ represents how many thousands of dollars the factory spends on quality control and $y$ represents how many thousands of dollars they spend on consulting. Find the average time if the amount spent on quality control varies from $\$ 0$ to $\$ 4000$ and the amount spent on consulting varies from $\$ 0$ to $\$ 2000$ . (Hint: Refer to Exercises $61-64 . )$

Time The time (in hours) that a branch of Amalgamated Entities needs to spend to meet the quota set by the main office can be approximated by

$$T(x, y)=x^{4}+16 y^{4}-32 x y+40$$

where $x$ represents how many thousands of dollars the factory spends on quality control and $y$ represents how many thousands of dollars they spend on consulting. Find the amount of money they should spend on quality control and on consulting to minimize the time spent, and find the minimum number of hours.

Time The time (in hours) that a branch of Amalgamated Entities needs to spend to meet the quota set by the main office car be appreximated by

$T(x, y)=x^{4}+16 y^{4}-32 x y+40$

where $x$ represents how many thousands of dollars the factory spends on quality control and $y$ represents how many thousands of dollars they spend on consulting. Find the amount of money they should spend on quality control and on consult-sing to minimize the time spent, and find the minimum number of hours.

Set up and solve Exercises 23–29 by the simplex method.

Profit The Muro Manufacturing Company makes two kinds of plasma screen television sets. It produces the Flexscan set that sells for $\$ 350$ profit and the Panoramic I that sells for $\$ 500$ profit. On the assembly line, the Flexscan requires 5 hours, and the Panoramic I takes 7 hours. The cabinet shop spends 1 hour on the cabinet for the Flexscan and 2 hours on the cabinet for the Panoramic I. Both sets require 4 hours for testing and packing. On a particular production run, the Muro Company has available 3600 work-hours on the assembly line, 900 work- hours in the cabinet shop, and 2600 work-hours in the testing and packing department. (See Exercise 10 in Section 3.3.)

a. How many sets of each type should it produce to make a maximum profit? What is the maximum profit?

b. Find the values of any nonzero slack variables and describe what they tell you about any unused time.

Use the four-step strategy to solve each problem. Use $x, y,$ and $z$ to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables.

A furniture company produces three types of desks: a children's model, an office model, and a deluxe model. Each desk is manufactured in three stages: cutting, construction, and finishing. The time requirements for each model and manufacturing stage are given in the following table.

CAN'T COPY THE FIGURE

Each week the company has available a maximum of 100 hours for cutting, 100 hours for construction, and 65 hours for finishing. If all available time must be used, how many of each type of desk should be produced each week?

The job performance of a new employee when learning a repetitive task (as on an assembly line) improves very quickly at first, then grows more slowly over time. This can be modeled by the function $P(t)=a+b \ln t$ where $a$ and $b$ are constants that depend on the type of task and the training of the employee.

The number of circuit boards an associate can assemble from its component parts depends on the length of time the associate has been working. This output is modeled by $B(t)=1+2.3 \mathrm{ln} t,$ where $B(t)$ is the number of boards assembled daily after working $t$ days. (a) How many boards is an employee completing after 9 days on the job? (b) How long will it take until the employee is able to complete 10 boards per day?

The job performance of a new employee when learning a repetitive task (as on an assembly line) improves very quickly at first, then grows more slowly over time. This can be modeled by the function $P(t)=a+b \ln t$ where $a$ and $b$ are constants that depend on the type of task and the training of the employee.

The number of toy planes an employee can assemble from its component parts depends on the length of time the employee has been working. This output is modeled by $P(t)=5.9+12.6 \mathrm{ln} t,$ where $P(t)$ is the number of planes assembled daily after working $t$ days. (a) How many planes is an employee making after 5 days on the job? (b) How many days until the employee is able to assemble 34 planes per day?

Social workers often use occupational test results when counseling their clients about employment options. The "learning curve" below shows that as a factory trainee assembled more chairs, the assembly time per chair generally decreased. If company standards required an average assembly time of 10 minutes or less, how many chairs did the trainee have to assemble before meeting company standards? (Notice that the graph is a model of exponential decay.)

Social workers often use occupational test results when counseling their clients about employment options. The “learning curve” below shows that as

a factory trainee assembled more chairs, the assembly time per chair generally decreased. If company standards required an average assembly time of 10 minutes or less, how many chairs did the trainee have to assemble before meeting company standards? (Notice that the graph is a model of exponential decay.)

(Graph can't copy)

Profit In Exercise 38 of Section $9.3,$ we saw that the profit (in thousands of dollars) that Aunt Mildred's Meta lworks earns from producing $x$ tons of steel and $y$ tons of aluminum can be approximated by

$P(x, y)=36 x y-x^{3}-8 y^{3}$

Find the average profit if the amount of steel produced varies from 0 to 8 tons, and the amount of aluminum produced varies from 0 to 4 tons. (Hint: Refer to Exercises $61-64 . )$