THE CASE: The economic decision of supplying the rice by a farmer depends on market price, ceteris paribus. The use of pesticides, irrigation and weather condition play pivotal role in the production of rice. Suppose that production of rice is increasing at the rate of the given equation below at time ‘t’ weeks from now. `\frac{dC(t)}{dt} = 0.6t^2 + 0.6t + 3` Requirement: 1. By how much will the quantity of bushel of rice be during the next 7 weeks if the market price remains fixed at $5 per bushel? 2. Calculate the value of crops for this production level. Note: Step wise calculations are required.

THE CASE:

The economic decision of supplying the rice by a farmer depends on market price, ceteris paribus. The use of pesticides, irrigation and weather condition play pivotal role in the production of rice. Suppose that production of rice is increasing at the rate of the given equation below at time ‘t’ weeks from now. 

`\frac{dC(t)}{dt} = 0.6t^2 + 0.6t + 3`

Requirement:

1. By how much will the quantity of bushel of rice be during the next 7 weeks if the market price remains fixed at $5 per bushel? 

2. Calculate the value of crops for this production level.
Note: Step wise calculations are required. 

Solution:

To solve this problem, we can integrate the given rate of production equation with respect to time 't' to find the cumulative production function. Once we have the cumulative production function, we can use it to answer the specific questions.

Given rate of production equation:

`dC(t)/dt =0.6t^2+0.6t+3`

To find the cumulative production function C(t), we integrate the given equation:

`C(t)=∫(0.6t^2+0.6t+3)dt`

Integrating each term separately:

`C(t)=0.2t^3+0.3t^2+3t + C1`

Now, we need to find the value of the constant C1. To do this, we can use the information that at time t=0, the production is zero:

C(0) = 0

0 = 0 + 0 + 0 + C1

C1 = 0

So, the cumulative production function is:

`C(t)=0.2t^3+0.3t^2+3t `

Now, let's proceed to answer the specific questions:

1. By how much will the quantity of bushel of rice be during the next 7 weeks?

To find the change in quantity over the next 7 weeks, we can subtract the cumulative production at the starting time from the cumulative production at the ending time:

`ΔC=C(t^2)−C(t1)`

In this case, let `t^1=0` (starting time) and `t^2=7` (7 weeks from now):

ΔC = C(7) – C(0)

`ΔC=(0.2(7)^3+0.3(7)^2+3(7))−(0.2(0)^3+0.3(0)^2+3(0))`

ΔC = 102.2

So, the quantity of bushel of rice will increase by 102.2 over the next 7 weeks.

2. Calculate the value of crops for this production level.

To find the value of crops, we multiply the quantity by the market price:

Value=Quantity×Price

Given that the market price is $5 per bushel, the value is:

Value=102.2×5=511

Therefore, the value of crops for this production level is $511.