In an economic enterprise, the total amount T that is produced is a function of the amount n of a given input used in the process of production. For example, the yield of a crop depends on the amount of fertilizer used, and the number of widgets manufactured depends on the number of workers. Because of the law of diminishing returns, a graph for T commonly has an inflection point followed by a maximum, so a cubic model may be appropriate. In this exercise we use the model shown below, with n measured in thousands of units of input and T measured in thousands of units of product.T = −2n3 + 3n2 + n(a) Make a graph of T as a function of n. Include values of n up to 1.5 thousand units. (b) Express using functional notation the amount produced if the input is 1.02 thousand units. (Round your answer to two decimal places.)T( )Calculate that value. (Round your answer to two decimal places.) thousand units(c) Find the approximate location of the inflection point. (Round your answers to one decimal place.)(n, T) = ( , )Explain what it means in practical terms.(d) What is the maximum amount produced? (Round your answer to two decimal places.)T = thousand units

(a) The graph of the function is as follows: (b) Here, T(1.02)=-2(1.02)3+3(1.02)2+1.02 =2.018784≈2.02 thousand units.(c) For the point of inflection the second derivative is zero. ⇒T '' (n)=0 ⇒T ''(n)=-12n+6 =0 ⇒n=12=0.5 Now, T(0.5)=-2(0.5)3+3(0.5)2+0.5=1. Thus, the point of inflection is ( 0.5, 1). The maximum rate of growth for T occurs at the point of inflection. Means T increases most when n is increased slightly at point of inflection. (d) The maximum amount occurs when T '(n)=0 ⇒-6n2+6n+1=0 ⇒n=-6±36+24-12=-6±60-12=-6±215-12=3±156 But n cannot be negative, so, n=3+156. T3+156=-23+1563+33+1562+3+156=2.07583≈2.08 thousand units.

Answered: In an economic enterprise, the total…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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In an economic enterprise, the total amount T that is produced is a function of the amount n of a given input used in the process of production. For example, the yield of a crop depends on the amount of fertilizer used, and the number of widgets manufactured depends on the number of workers. Because of the law of diminishing returns, a graph for T commonly has an inflection point followed by a maximum, so a cubic model may be appropriate. In this exercise we use the model shown below, with n measured in thousands of units of input and T measured in thousands of units of product.

T = −2n³ + 3n² + n

(a) Make a graph of T as a function of n. Include values of n up to 1.5 thousand units.

(b) Express using functional notation the amount produced if the input is 1.02 thousand units. (Round your answer to two decimal places.)
T( )

Calculate that value. (Round your answer to two decimal places.)
thousand units

(c) Find the approximate location of the inflection point. (Round your answers to one decimal place.)

(n, T)

( , )

Explain what it means in practical terms.

(d) What is the maximum amount produced? (Round your answer to two decimal places.)
T = thousand units

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