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Umnica [9.8K]
3 years ago
15

You have been contracted to replace a countertop in a kitchen. You will get paid $1,000 to do the job. You can complete the job

in three days working 8 hours a day. You will need to pay the cost of the materials, which is $525, with the $1,000 that you receive to do the job. How much will you be paid per hour for your time?
Mathematics
1 answer:
ElenaW [278]3 years ago
6 0

Answer:

$19.79 cents (rounded to the nearest cent)

Step-by-step explanation:

first subtract the amount needed for materials from your total pay

(1000 - 525 = 475)

take 475 and divide that among the hours you work.

475 / 24 = 19.7916666

round that number to the closest money value

19.79

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f_x=4x-4=0\implies x=1
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\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}4&0\\0&6\end{bmatrix}

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To find the extrema (if any) along the boundary, parameterize it by x=4\cos t and y=4\sin t, with 0\le t. On the boundary, we have

f(x(t),y(t))=F(t)=2(4\cos t)^2+3(4\sin t)^2-4(4\cos t)-5=32\cos^2t+48\sin^2t-16\cos t-5
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Find the critical points along the boundary:

F'(t)=16\sin t+16\sin2t=16\sin t+32\sin t\cos t=16\sin t(1+2\cos t)=0
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Respectively, plugging these values into F(t) gives 11, 47, 43, and 47. We omit the first and third, as we can see the absolute extrema occur when F(t)=47.

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t=\dfrac{2\pi}3\implies\begin{cases}x=4\cos t=-2\\y=4\sin t=2\sqrt3\end{cases}

t=\dfrac{4\pi}3\implies\begin{cases}x=4\cos t=-2\\y=4\sin t=-2\sqrt3\end{cases}

so f(x,y) has two absolute maxima at (x,y)=(-2,\pm2\sqrt3) with the same value of 47.
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