Chapter 3.5, Problem 37PS

Solution

$\left(a\right)$

To write an equation for the speed $v$ of sound waves in air as a function of temperature $C$ in degree Celsius. Kelvin temperature is related to the Celsius temperature by the formula $K=273.15+C$ .

  $v=331.5\sqrt{\frac{C+273.15}{273.15}}$

Given information:

The speed $v$ in (meter per second) of sound wave in air depend on the temperature $k$ (in kelvins) and can be modelled by the following equation:

  $v=331.5\sqrt{\frac{k}{273.15}},k\ge 0$

Explanation:

Substitute $C+273.15$ for $k$ in the given equation:

  $v=331.5\sqrt{\frac{C+273.15}{273.15}}$

The above equation gives the speed $v$ of sound waves in air as a function of temperature $C$ in degree Celsius.

$\left(b\right)$

To write the domain and the range for the function from part $\left(a\right)$ .

The domain for the function is $x\ge 0$ and the range for the function is $y\ge 0$ .

Given information:

The speed $v$ in (meter per second) of sound wave in air depend on the temperature $k$ (in kelvins) and can be modelled by the following equation:

  $v=331.5\sqrt{\frac{k}{273.15}},k\ge 0$

Explanation:

The equation from the part $\left(a\right)$ is:

  $v=331.5\sqrt{\frac{C+273.15}{273.15}}$

The above equation gives the speed $v$ of sound waves in air as a function of temperature $C$ in degree Celsius.

Since $k\ge 0$ , it follows that:

  $C+273.15\ge 0$

The radicand of the square root must be non-negative:

Therefore, the domain for the function is $x\ge 0$ and the range for the function is $y\ge 0$ .