   Chapter 7.5, Problem 26QY ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model. The total cost (in thousands of dollars) for producing x freestanding stoves and y fireplace-insert stoves can be modeled by C ( x , y ) = 1 16 x 2 + y 2 − 10 x − 10 y + 820 Find the values of x and y that minimize the total cost. What is the minimum cost?

To determine

To calculate: To find the value of the variable x and y so that the total cost will minimize and

also find the minimum cost.

Explanation

Given Information:

The company manufactures two types of wood baring stoves: a trading model and fine place

insert model. The total cost (in thousand dollars).

For producing x free standing stoves and y fine place insert can be modelled by,

c(x,y)=116x2+y210x40y+820

Formula used:

Step 1: Write the given expression of the model.

Step 2: Find the partial derivatives of first order of the function.

Step 3: To find the critical value equating them with zero.

Step 4: Find the double derivatives and find the values of them at the critical point.

Step 5: let (a,b) be the critical point and,

d=cyy(a,b)cxx(a,b)[cxy(a,b)]2

Then if d>0 and cxx(a,b)>0 then c has relative maximum at (a,b).

Step 6: Putting the value of that variables in the original equation find the extreme cost.

Calculation:

The given model is,

c(x,y)=116x2+y210x40y+820

Now, to find the critical points equating the partial derivatives of first order with zero.

cx(x,y)=0cy(x,y)=0

Now,

cx(x,y)=1162x10cx(x,y)=x810cy(x,y)=2y40

Equating the partial derivative 0,

x810=02y40=0

That is,

x810=0x=80

And

2y40=0y=20

So the critical point is (80,20)

Now, find the double partial derivatives with respect to x,

cxx(x,y)=ddx(cx(x,y))=18</

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