Administrators at MG hospital believe that the hospital's expenditures E(b) measures in dollars, are a function of how many beds (B) are in use with; E(b)=1400 + (B+1)^2. On the other hand, the number of beds is a function of time t, measured in days and it is estimated that: B(t)= 20sin(t/10) + 50. At what rate are the expenditures decreasing when t =100?

Question

ganeshie8 · Answer

use chain rule

ganeshie8 · Answer

$\dfrac{dE}{dt}  =  \dfrac{dE}{dB}  \times \dfrac{dB}{dt}$

anonymous · Answer

what would dE/dB be?

ganeshie8 · Answer

$E=1400 + (B+1)^2 $

differentiate with respect to B

anonymous · Answer

so 2(B+1) * 2cos(t/10) ?

ganeshie8 · Answer

looks good,
plugin B value[B= 20sin(t/10) + 50]

and evaluate at t = 100

anonymous · Answer

I'm getting -135 when I should be getting a +135

ganeshie8 · Answer

$\dfrac{dE}{dt}  =  \dfrac{dE}{dB}  \times \dfrac{dB}{dt} $

$~~~~~ =  2(B+1)\times 2\cos(\frac{t}{10})$

$~~~~~ =  2(20\sin(t/10) + 50 +1)\times 2\cos(\frac{t}{10})$

ganeshie8 · Answer

$\dfrac{dE}{dt}=  2(20\sin(t/10) + 50 +1)\times 2\cos(\frac{t}{10})$


$\dfrac{dE}{dt}\Bigg |_{t=100 }    = -135$

anonymous · Answer

OH cause its decreasing so thats why its negative right?

ganeshie8 · Answer

that means expenditures are DECREASING at a rate of 135

ganeshie8 · Answer

you got it !

ganeshie8 · Answer

On number line :

"-135"   is same as saying "left +135 units"

anonymous · Answer

ok thanks!

ganeshie8 · Answer

np :)