Question

Find the rate of change of total​ revenue, cost, and profit with respect to time. Assume that​ R(x) and​ C(x) are in dollars.
R(x)= 2x, C(x)= 0.01x² + 0.4x + 30, when x=25 and dx/dt=7 units per day.
1. Find the rate of change of total revenue per day.
2. Find the rate of change of total cost per day.
3. Find the rate of change of total profit per day.

Answer 1

Answer:

1.- dR/dt   = 14 $/day

2.- dC/dt =  4.55 $/day

3.- dP/dt = 7,7 $/day

Step-by-step explanation:

By definition Profit is equal to Revenue minus total costs then.

We have

R(x) = 2*x

C(x) = 0,01*x²  + 0,4*x  + 30

Then Profit  P(x)  =  2*x  -  0,01*x²  - 0,4*x  - 30

P(x) = - 0.01*x²  +  1.6*x  - 30

1.- Find rate of change total revenue per day, when

x = 25  and dx/dt = 7 u/day

R(x) = 2*x

dR/dt   = 2*dx/dt      ⇒   dR/dt   = 2* 7

dR/dt   = 14 $/day

2.-

C(x) = 0,01*x²  + 0,4*x  + 30

dC/dt = 0,01*x* dx/dt  + 0,4*dx/dt

dC/dt = 0,01*25*7  +  0.4*7

dC/dt = 1.75 + 2.8

dC/dt =  4.55 $/day

3.-

P(x) = - 0.01*x²  +  1.6*x  - 30

dP/dt = - 2*0,01*x*dx/dt  + 1.6*dx/dt

dP/dt = - 2*0,01*25*7  + 1.6*7

dP/dt = - 3,5 + 11.2

dP/dt = 7,7 $/day

   

Answer 2

Answer:

1. The rate of change of total Revenue is $14 per day.
2. The rate of change of total Cost is $4.55 per day.
3. The rate of change of total Profit is $7.7 per day.

Step-by-step explanation:

Given information

$R(x)=2x\\C(x)=0.01x^2+0.4x+30$

When,

$x=25\\dx/dt=7$ units per day

As we know that

Profit = Revenue - Cost

Then, the Profit,

$P(x)=2x-0.01x^2+0.4x-30\\P(x)=-0.01x^2+1.6x-30$

Now, The total Revenue per day

$R(x)=2*x$

By differentiating both side with respect to $t$

$\\dR/dt=2*dx/dt\\dR/dt=2*7\\dR/dt=14$

Hence the rate of change of total revenue is $14 per day.

Similarly,

The total cost per day

$C(x)=0.01x^2+0.4x+30$

By differentiating both side with respect to $t$

$dC/dT=0.01*x*dx/dt+0.4*dx/dt\\dC/dt=0.01*25*7+0.4*7\\dC/dt=1.75+2.8\\dC/dt=4.55$

Hence the rate of change of total cost is $4.55 per day

And the total Profit per day

$P(x)=-0.01x^2+1.6x-30$

By differentiating both side with respect to $t$

$dP/dt=2*(-0.01)*x*dx/dt+1.6*dx/dt\\dP/dt=-2*(-001)*25*7+1.6*7\\dP/dt=-3.5+11.2\\dP/dt=7.7$

Hence the rate of change of total Profit is $7.7 per day

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