Chapter 4, Problem 15CT

To determine

To find the length of the minute hand of a clock.

Answer to Problem 15CT

The length of the minute hand of a clock is 12 cm.

Explanation of Solution

Given information:

The minute hand of a clock is 3 cm longer than the hour hand and the distance between the tips of the hands at nine o’clock is 15 cm.

  

Let, the length of the hour hand of a clock, $BC$$=$$x$ cm.

  $\therefore$ Length of the minute hand of a clock, $AB$$=\left(x+3\right)$ cm.

Pythagoras theorem:

(Hypotenuse)2 = (Base)2 + (Height)2

Now, according to the Pythagoras theorem-

In $△ABC$ , $A{C}^2=B{C}^2+A{B}^2$

  $\begin{array}{l}{\left(15\right)}^2={\left(x\right)}^2+{\left(x+3\right)}^2\\ 225=x^2+x^2+6x+9\\ 225=2x^2+6x+9\\ 0=2x^2+6x+9-225\\ 2x^2+6x-216=0\\ 2\cdot \left(x^2+3x-108\right)=0\\ x^2+3x-108=0\\ x^2+12x-9x-108=0\\ x\cdot \left(x+12\right)-9\cdot \left(x+12\right)=0\\ \left(x+12\right)\cdot \left(x-9\right)=0\\ \\ \therefore x+12=0\therefore x-9=0\\ x=-12x=9\end{array}$

Since $x$ represents a length, so it cannot be negative.

  $\therefore$$x=-12$ cannot be the length of the hour hand of a clock.

Here, the only reasonable solution is $x=9$ .

So, the length of the hour hand of a clock is 9 cm.

  $\therefore$ The length of the minute hand of a clock $=x+3=9+3=12$ cm.

Hence,

The length of the minute hand of a clock is 12 cm.