Interpreting calculus 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

Part A: Interpret f(20) = 11,500

When 20 kg of fertilizer is used on an average plot of land in Kenya, the value of corn produced is 11,500 Kenyan shillings.

Part B: Interpret f'(20) = 350

When 20 kg of fertilizer is being used, the value of corn production is increasing at a rate of approximately 350 Kenyan shillings for each additional kilogram of fertilizer.

Remember, the derivative tells us how quickly something is changing!

Part C: Estimate f(21)

We know that using 20 kg of fertilizer produces corn worth 11,500 shillings.

From Part B, we also know that adding another kilogram of fertilizer increases the value by approximately 350 shillings.

So, we simply add:

f(21) ≈ 11,500 + 350(1)

f(21) ≈ 11,850

Final Answer: 11,850 Kenyan shillings

This means that using 21 kg of fertilizer would produce corn worth approximately 11,850 Kenyan shillings.

The key idea is that the derivative gives us an estimate of how much the value changes when we increase the fertilizer by a small amount. Since we're only increasing it by 1 kg, we can use the derivative to make a reasonable estimate!