BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 2, Problem 31P

(a)

To determine

To show: The total energy used by the bird is E(x)=14x2+25+10(12x).

Expert Solution

Explanation of Solution

Formula used:

energyused=energyper mile ×milesflown (1)

Calculation:

The energy of the bird requires 10kcal/mi to fly over the land and 14kcal/mi to fly over the water.

All the distances related to the bird are given in the figure below,

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 2, Problem 31P , additional homework tip  1

Let x be distance between point B and C.

In ΔABC,

Use the Pythagoras theorem to find the length AC.

AC2=AB2+BC2[Hypothesis2=base2+perpendicular2]=(5)2+x2AC=25+x2

The length of the side BD is 12mi and the length of the side BC is x mi.

Then, the length of the side CD is,

CD=BDBC=12x

The bird flies from the point C to the point D on the land with the energy 10kcal/mi.

Energyforonemile=10kcal (2)

To find the energy for (12x)mile, multiply by 12x both side of the equation (2),

Energyfor(12x)mile=10(12x)kcal

The bird flies from the point A to the point C on the water with the energy 14kcal/mi.

Energyforonemile=14kcal (3)

To find the energy for 25+x2mile, multiply 25+x2 on both sides of the equation (3),

Energyfor25+x2mile=1425+x2kcal

Then the total energy is,

TotalEnergy=Energyfor(12x)mile+Energyfor25+x2mile (4)

Substitute 10(12x)kcal for Energyfor12xmile and 1425+x2kcal for Energyfor25+x2mile in the equation (4),

TotalEnergy=Energyfor(12x)mile+Energyfor25+x2mileE(x)=10(12x)kcal+1425+x2kcal=(10(12x)+1425+x2)kcal

Thus, the total energy is E(x)=14x2+25+10(12x).

(b)

To determine

To find: The path that minimizes the energy expenditure.

Expert Solution

Answer to Problem 31P

The bird flies till the point C on shoreline which is 5.013 miles from point B and then flies along the shoreline.

Explanation of Solution

From the part (a), the total energy is E(x)=14x2+25+10(12x).

To find the minimum energy sketch the graph of E(x)=14x2+25+10(12x).

The function contains the variable x is the length of the side BC.

The local minimum value of the function is the least finite value where the value of the function at the any number is less than to the original function.

The condition for local minimum is,

f(a)f(x)

Using online graphing calculator, sketch the graph of the function E(x)=14x2+25+10(12x) as shown in the figure below.

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 2, Problem 31P , additional homework tip  2

From the above figure, it can be observed that the least peak occurs at the point (5.103,168.99).

Then, the minimum energy expenditure is 168.99 at x=5.103.

Thus, the point C is 5.103 mi away from point B.

Therefore, the bird flies till the point C on shoreline which is 5.013 miles from point B and then flies along the shoreline.

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