The amount of trash G (t) , in tons per year, produced by a town is projected to continue growing according to the formula
dG/d t = 80 + 4t
. Based on this projection, calculate the total amount of trash that will be produced over the next 5 years, with t = 0 corresponding to the present time.
if dG/d t = 80 + 4t , then
G = 80t + 2t^2 + c
when t = 0, G = c
when t = 5, G = 80(5) + 2(25) + c
trash produced = 400 + 50 + c - c = 450
or
∫ (80 + 4t) dt from t = 0 to 5
= [80t + 2t^2] from 0 to 5
= ( 80(5) + 2(25) - 0)
= 450
Well, it looks like the rate at which the trash is growing is given by the derivative of the trash function, which is 80 + 4t. To find the total amount of trash produced over the next 5 years, we'll need to integrate this function from t = 0 to t = 5.
So, let's integrate 80 + 4t with respect to t:
∫(80 + 4t) dt = 80t + 2t^2 + C
Now, we can evaluate this integral from t = 0 to t = 5:
[80(5) + 2(5^2) + C] - [80(0) + 2(0^2) + C]
Simplifying this expression, we get:
400 + 50 + C - 0 - 0 - C
The C terms cancel out, leaving us with:
450
Therefore, the total amount of trash that will be produced over the next 5 years is 450 tons.
That's a lot of trash! I hope they recycle!
To calculate the total amount of trash that will be produced over the next 5 years, we need to integrate the given formula over the time interval from t = 0 to t = 5.
The formula is given as:
dG/dt = 80 + 4t
Integrating both sides with respect to t will give us the total amount of trash produced, G(t):
∫ dG = ∫ (80 + 4t) dt
Integrating the right side:
G(t) = ∫ (80 + 4t) dt
= 80t + 2t^2 + C
To evaluate the constant of integration, C, we need to know the initial amount of trash at t = 0. Assuming that there is no trash at the present time, G(0) = 0.
Plugging in G(0) = 0, we have:
0 = 80(0) + 2(0)^2 + C
0 = 0 + 0 + C
C = 0
Therefore, the equation for the total amount of trash produced is:
G(t) = 80t + 2t^2
Now, let's calculate the total amount of trash produced over the next 5 years, starting from t = 0.
G(5) = 80(5) + 2(5)^2
= 400 + 2(25)
= 400 + 50
= 450
Therefore, the total amount of trash that will be produced over the next 5 years is 450 tons.
To find the total amount of trash that will be produced over the next 5 years, we need to integrate the rate of change equation with respect to time (t) and then evaluate it from t = 0 to t = 5.
Given that the rate of change equation is:
dG/dt = 80 + 4t
To find the total amount of trash (G) over the next 5 years, we integrate the equation:
∫ (dG/dt) dt = ∫ (80 + 4t) dt
Integrating with respect to t, we get:
∫ dG = ∫ (80 + 4t) dt
Now, integrating both sides:
G = ∫ (80 + 4t) dt
To integrate, we use the power rule of integration:
G = ∫80 dt + ∫4t dt
Integrating each term separately:
G = 80∫ dt + 4∫t dt
The integral of dt is t, and the integral of t dt is (1/2)t^2. Substituting these values:
G = 80t + 2t^2 + C
Here, C is the constant of integration.
Now, we need to evaluate the equation from t = 0 to t = 5:
G(5) - G(0) = (80(5) + 2(5^2) + C) - (80(0) + 2(0^2) + C)
Simplifying:
G(5) - G(0) = 400 + 2(25)
G(5) - G(0) = 400 + 50
G(5) - G(0) = 450
So, the total amount of trash that will be produced over the next 5 years is 450 tons.