Chapter 3.1, Problem 63PS

Solution

The value of k and the speed of the fan if it uses $1.5$ horsepower.

  $k=\frac{1.2}{4913000000}$

  $1831$ revolutions per minute

Given:

The power p in horsepower used by a fan with rotational speed s in revolutions per minute can be modeled by the formula $p=k{s}^3$ . A certain fan uses $1.2$ horsepower when its speed is 1700 revolutions per minute.

Concept Used:

• The even inter root of a number is defined for positive values only that are greater than or equal to 0.
• The odd integer root of all real numbers is defined.
• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• The equivalent expression in rational exponent for any nth radical is given by, $\sqrt[n]{a}=a^{\frac{1}{n}}$ .

Calculation:

In order to find the value of k , substitute $p=1.2,\text{ and }s=1700$ into the given formula and simplify further as shown below

  $\begin{array}{c}p=k{s}^3\\ 1.2=k{\left(1700\right)}^3\\ 1.2=4913000000k\\ k=\frac{1.2}{4913000000}\end{array}$

So, the value of k is $k=\frac{1.2}{4913000000}$ .

Now to find the speed of the fan if it uses 1.5 horsepower, substitute $k=\frac{1.2}{4913000000}$ and $p=1.5$ into the given formula and simplify further as shown below

  $\begin{array}{c}p=k{s}^3\\ 1.5=\frac{1.2}{4913000000}\cdot s^3\\ s^3=\frac{4913000000\times 1.5}{1.2}\\ s=\sqrt[3]{\frac{4913000000\times 1.5}{1.2}}\\ s\approx 1831\end{array}$

Thus, the speed of the fan is about $1831$ revolutions per minute.