Chapter 2.2, Problem 42E

a.

To determine

To find: The cost of measuring the temperature inside the sun 10,000 below the surface.

Answer to Problem 42E

The cost of measuring the temperature 10,000 km below the surface is $ 1 000,000

(One million dollars).

Explanation of Solution

Given:

Inside the sun the temperature increases by approximately $\text{1}\text{million}\text{C}\text{=1}{\text{0}}^{\text{6}}\text{C}$ for every 10,000 km below the surface. A probe (temperature measuring device) that can handle temperature of x million degree costs $x^{\text{3}}$ million

Concept used:

The cost of measuring temperature is $C={\left\{x(\text{temperature}\text{in}\text{C})\right\}}^3\text{dollars}$ .

Calculation:

The temperature of the surface of the sun is $5505C$ , and below the surface it increases $\text{1}\text{million}\text{C}$ for every 10,000 km.

Therefore, the temperature athkm below the surface is

  $x(\text{in}\text{C})=\text{5505}\text{C}+\left(\frac{h}{10,000}\right)\text{1}{\text{0}}^{\text{6}}\text{C}$   ...(1)

The cost of measuring the temperature of xmillion degree $\left(x\times {10}^6\text{C}\right)$ ) is

  $C=x^3\text{dollars}$  ...(2)

The temperature at 10,000m down below the surface is

  $x(\text{in}\text{C})=\text{5505}\text{C}+\left(\frac{10,000}{10,000}\right)\times \text{1}{\text{0}}^{\text{6}}\text{C}$

  $\Rightarrow x\left(\text{in}\text{C}\right)\text{=}\left(0.005505+1\right)\times \text{1}{\text{0}}^{\text{6}}\text{C}\text{=}\left(1.005505\right)\times \text{1}{\text{0}}^{\text{6}}\text{C}\approx 1{\text{0}}^{\text{6}}\text{C}\text{=1 }\text{million}\text{C}$

Therefore, the cost of measuring the temperature 10,000 km down below the surface is

  $C={\left(1\right)}^3=1\text{million}\text{dollars=\$1000,000}$ .

Conclusion:

The cost of measuring the temperature 10,000 km below the surface is $ 1 000,000

(One million dollars).

b.

To determine

To find: The cost of measuring the temperature inside the sun 100,000 below the surface.

Answer to Problem 42E

The cost of measuring the temperature 100,000 km below the surface is $\text{ \$1}{\text{0}}^{\text{9}}$ (1 billion dollars).

Explanation of Solution

Given:

Inside the sun the temperature increases by approximately $\text{1}\text{million}\text{C}\text{=1}{\text{0}}^{\text{6}}\text{C}$ for every 10,000 km below the surface. A probe (temperature measuring device) that can handle temperature of x million degree costs $x^{\text{3}}$ million

Concept used:

The cost of measuring temperature is $C={\left\{x(\text{temperature}\text{in}\text{C})\right\}}^3\text{dollars}$ .

Calculation:

The temperature of the surface of the sun is $5505C$ , and below the surface it increases $\text{1}\text{million}\text{C}$ for every 10,000 km.

Therefore, the temperature athkm below the surface is

  $x(\text{in}\text{C})=\text{5505}\text{C}+\left(\frac{h}{10,000}\right)\text{1}{\text{0}}^{\text{6}}\text{C}$   ...(1)

The cost of measuring the temperature of x million degree $\left(x\times {10}^6\text{C}\right)$ ) is

  $C=x^3\text{dollars}$  ...(2)

The temperature at 100,000m down below the surface is

  $x(\text{in}\text{C})=\text{5505}\text{C}+\left(\frac{100,000}{10,000}\right)\times \text{1}{\text{0}}^{\text{6}}\text{C}$

  $\Rightarrow x\left(\text{in}\text{C}\right)\text{=}\left(0.005505+10\right)\times \text{1}{\text{0}}^{\text{6}}\text{C}\text{=}\left(10.005505\right)\times \text{1}{\text{0}}^{\text{6}}\text{C}\approx 10\times 1{\text{0}}^{\text{6}}\text{C}\text{=10}\text{million}\text{C}$

Therefore, the cost of measuring the temperature 100,000 km down below the surface is

  $C={\left(10\right)}^3=1000\text{million}\text{dollars=1billion dollars}$ .

Conclusion:

The cost of measuring the temperature 100,000 km below the surface is $\text{ \$1}{\text{0}}^{\text{9}}$ (One billion dollars).

c.

To determine

To find: The cost of measuring the temperature inside the sun 200,000 below the surface.

Answer to Problem 42E

The cost of measuring the temperature 200,000 km below the surface is $\text{ \$8}\times \text{1}{\text{0}}^{\text{9}}$ (8 billion dollars).

Explanation of Solution

Given:

Inside the sun the temperature increases by approximately $\text{1}\text{million}\text{C}\text{=1}{\text{0}}^{\text{6}}\text{C}$ for every 10,000 km below the surface. A probe (temperature measuring device) that can handle temperature of x million degree costs $x^{\text{3}}$ million

Concept used:

The cost of measuring temperature is $C={\left\{x(\text{temperature}\text{in}\text{C})\right\}}^3\text{dollars}$ .

Calculation:

The temperature of the surface of the sun is $5505C$ , and below the surface it increases $\text{1}\text{million}\text{C}$ for every 10,000 km.

Therefore, the temperature athkm below the surface is

  $x(\text{in}\text{C})=\text{5505}\text{C}+\left(\frac{h}{10,000}\right)\text{1}{\text{0}}^{\text{6}}\text{C}$   ...(1)

The cost of measuring the temperature of x million degree $\left(x\times {10}^6\text{C}\right)$ ) is

  $C=x^3\text{dollars}$  ...(2)

The temperature at 200,000m down below the surface is

  $x(\text{in}\text{C})=\text{5505}\text{C}+\left(\frac{200,000}{10,000}\right)\times \text{1}{\text{0}}^{\text{6}}\text{C}$

  $\Rightarrow x\left(\text{in}\text{C}\right)\text{=}\left(0.005505+20\right)\times \text{1}{\text{0}}^{\text{6}}\text{C}\text{=}\left(20.005505\right)\times \text{1}{\text{0}}^{\text{6}}\text{C}\approx 20\times 1{\text{0}}^{\text{6}}\text{C}\text{=20 }\text{million}\text{C}$

Therefore, the cost of measuring the temperature 200,000 km down below the surface is

  $C={\left(20\right)}^3=8000\text{million}\text{dollars=8}\text{billion dollars}$ .

Conclusion:

The cost of measuring the temperature 200,000 km below the surface is $\text{ \$8}\times \text{1}{\text{0}}^{\text{9}}$ (8 billion dollars).