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 $61-64$ .

78.4 $\mathrm{hr}$

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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.

Georgia Southern University