Answer:
1300
Step-by-step explanation:
We are given
x = 35
[tex]\frac{dp}{dt}=-0.05[/tex] [rate of change of price with time]
We want to find [tex]\frac{dx}{dt}[/tex], which is the rate at which supply is changing.
First, we need to implicitly differentiate the equation given:
[tex]625p^2-x^2=100\\1250p\frac{dp}{dt}-2x\frac{dx}{dt}=0[/tex]
Here, we have what we need except for p. We find p when x = 35:
[tex]625p^2 - x^2 =100\\625p^2 - (35)^2 =100\\625p^2=1325\\p=1.46[/tex]
Now, find dx/dt:
[tex]1250(1.46)(-0.05)-2(35)\frac{dx}{dt}=0\\-91.25-70\frac{dx}{dt}=0\\\frac{dx}{dt}=-1.30[/tex]
In thousands, 1.30 * 1000 = 1300 cartons per week decrease
