calc word problem

Question

Fertilizers can improve agriculture. A research of corn grown in Kenya found that the value, V=f(x), in Kenyan shillings of corn production from an average plot of land is a function of quantity, x, of fertilizer used in kg.

 

a.Using a complete sentence, interpret the statement f(20)=11,500.

 

b.Using a complete sentence, interpret the statement f'(20)=350.

 

c.Use part (b) to estimate f(21). 

Answer 1

To solve this problem, we first define what we know. The problem states that x is the quantity of fertilizer used in kilograms and V is the production number of Kenyan shillings of corn. Therefore, the amount of fertilizer placed on the land gives you more or less shillings of corn.

 

Part A: If f(x) is replaced with f(20) then that means x is 20. Therefore, we know that 20kg of fertilizer has been placed on the field. The result is that 11,500 shillings of corn are produced since f(20) = 11,500 and 11,500 is your "V."

Complete Sentence: If 20kg of fertilizer is planted on a plot of land, then you produce 11,500 shillings of corn.

 

Part B: If f(x) is the amount of shillings of corn produced, then f'(x) is the rate at which it is produced and is defined as the derivative of f(x). Therefore you could say the below sentence:

The rate of shillings of corn being produced from 20kg of fertilizer is 350.

 

Part C: If we take an undefined integral of the rate (the antiderivative of f'(x)) then we should get f(x) = 350x + C

If we say f(20) = 11,500 then substituting 20 for x, we get 11,500 = 350 (20) + C

Solve for C and we get C = 4,500

Therefore we have f(x) = 350x + 4,500

If we substitute 21 for x then f(21) = 350(21) + 4,500

Therefore we have our estimate that f(21) = 11,850 shillings of corn

 

If you are curious about how I got certain quantities or to certain conclusions, please send me a message :)

Answer 2

(a) f(20) = 11500

20 kg of fertilizer will result in 11500 Kenyan shillings of corn production


(b) f'(20) = 350

The value of the corn production increases by 350 Kenyan shillings for 20 kg of fertilizer.


(c) f(21) - f(20)
--------------- = f'(20) <--- this is the best we got
21-20

f(21) - 11500 = 350 <--- denominator is 1

f(21) = 11850