Chapter 5, Problem 76RE

Solution

To find: The name of the person that records the fake data and write method to solve.

The faked data is recorded by Pat. Form an expression for $2x$ in terms of $\theta$ to compare the recorded data.

Given information: The figure is given as follows:

  

Figure (1)

Formula used: The law of cosine for a triangle $ABC$

is given as follows:

  $a^2=b^2+c^2-2bc\cos \theta$

Calculation:

Apply the law of cosine in $\Delta ABC$ shown in the given figure.

  $\begin{array}{c}{\left(24\right)}^2=x^2+x^2-2\left(x\right)\left(x\right)\cos \theta \\ 576=2x^2-2x^2\cos \theta \\ 576=2x^2\left(1-\cos \theta \right)\\ \frac{576}{2\left(1-\cos \theta \right)}=x^2\end{array}$

Further simplify the above equation.

  $\begin{array}{c}\frac{288}{\left(1-\cos \theta \right)}=x^2\\ x=\pm \sqrt{\frac{288}{1-\cos \theta }}\end{array}$

Consider the positive value of $x$, as the distance cannot be negative.

  $\begin{array}{c}x=\sqrt{\frac{288}{1-\cos \theta }}\\ 2x=2\sqrt{\frac{288}{1-\cos \theta }}\end{array}$

Substitute different values of $\theta$ in the above equation and draw a table for the values of $2x$ .

$\theta$	$2x$
$160°$	$24.37$
$140°$	$25.54$
$120°$	$27.71$
$100°$	$31.32$
$80°$	$37.34$
$60°$	$48$
$40°$	$70.17$
$20°$	$138.21$

From the above table, it can be observed that the data matches with the values recorded by the Carmen. So, Carmen recorded the correct data.

Thus, the faked data is recorded by Pat. Form an expression for $2x$ in terms of $\theta$ to compare the recorded data.