Chapter 9.7, Problem 38E

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Chapter
Section

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

# Production Suppose that the production of x items of a new line of products is given by x = 200 [ ( t + 10 ) − 400 ( t + 40 ) − 1 ] where t is the number of weeks the line has been in production. Find the rate of production, dx/dt.

To determine

To calculate: The rate of production, dxdt, of x items of a new line of products which has the production function x=200[(t+10)400(t+40)1].

Explanation

Given Information:

The production of x items for a new line of product is represented by the function, x=200[(t+10)âˆ’400(t+40)âˆ’1]. Here, t is the number of weeks the line has been in production.

Formula used:

According to the power rule, if f(x)=xn, then,

fâ€²(x)=nxnâˆ’1

According to the property of differentiation, if a function is of the form, g(x)=cf(x), then,

gâ€²(x)=cfâ€²(x)

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

fâ€²(x)=uâ€²(x)+vâ€²(x)

The derivative of a constant value, k, is

ddx(k)=0

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nunâˆ’1dudx

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nunâˆ’1dudx

Calculation:

Consider the provided production function,

x=200[(t+10)âˆ’400(t+40)âˆ’1]

The rate of change of production is found by differentiating the revenue function with respect to t. Consider (t+40) to be u,

x=200[(t+10)âˆ’400uâˆ’1]

Differentiate both sides with respect to t,

dxdt=ddt(200[(t+10)âˆ’400uâˆ’1])=200(ddt[(t+10)âˆ’400uâˆ’1])

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