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Sagot :
72 units of wheat must be planted by to farmer for maximum yield and In order to get maximum profit the farmer should plant, 61 units of wheat on land.
Give a brief account on derivative.
In mathematics, the derivative of a function of a real variable measures how sensitive the function's value (or output value) is to variations in its argument (input value). A measure of how quickly an object's position changes over time is its velocity, which is the derivative of that object's position with respect to time.
The derivative has a wide range of real-world applications. A function's derivative can be used to find the maxima and minima of the function. Maximizing production, earnings, and minimizing losses are helpful in real life.
To solve the question :
Given:
Planted units of wheat = x
Yield of the farmland Y = [tex]12x- \frac{x^{2} }{12}[/tex]
Cost of production = $110/unit
Revenue from harvest = $60/unit
(a) For maximum yield,
[tex]\frac{dY}{dx} = 0[/tex]
[tex]0= 12x- \frac{x^{2} }{12}[/tex]
[tex]x=72[/tex]
Thus, 72 units of wheat must be planted by to farmer for maximum yield.
(b) Profit (P) = Revenue - Production cost
[tex](60( 12x- \frac{x^{2} }{12})) -(110x)[/tex]
[tex]P=610 x-5x2[/tex]
[tex]\frac{dP}{dx} = 0[/tex]
[tex]610-10x=0[/tex]
[tex]x[/tex] = 61 units
In order to get maximum profit the farmer should plant, [tex]x[/tex] = 61 units of wheat on land.
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The complete question is mentioned below :
Solve the problems below that pertain to a farmer planting wheat.
a. If a farmer plants x units of wheat in his field,
0≤ [tex]x[/tex] ≤ 144, the yield will be [tex]12x- \frac{x^{2} }{12}[/tex] units. How much wheat should he plant for the maximum yield?
_____ units of wheat
b. In the problem above, it costs the farmer $110 for each unit of wheat he plants, and he is able to sell each unit he harvests for $60. How much should he plant to maximize his profit?
_____ units of wheat
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