Chapter 12.6, Problem 12P

To determine

To find: number of students went for the computer museum.

Answer to Problem 12P

The number of students went for the computer museum is $19$ .

Explanation of Solution

Given information:

Total cost for the trip was $\$240$ and total number of students not decided to go for the trip was $4$ .

Also the increase in the cost by $\$2$ .

Formula:

The quadratic equation $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

Calculation:

Consider that the student went for the trip was $x$ .

Total cost for the trip was $\$240$ .

Then the cost for $1$ student will be $\frac{240}{x}$ .

Since as mentioned above that total number of students not decided to go for the trip was $4$ .

Therefore the number of student went for the trip was $\left(x-4\right)$ .

Then cost was increased by $\$2$ .

Therefore total cost for the trip will be $\frac{240}{x}+2$ .

Cost for $1$ student will be $\frac{240}{x-4}$ .

Therefore $\frac{240}{x}+2=\frac{240}{x-4}$ .

Therefore $\begin{array}{l}\frac{240}{x}+2=\frac{240}{x-4}\Rightarrow \frac{240}{x-4}-\frac{240}{x}=2\Rightarrow \frac{240\left(x\right)-240\left(x-4\right)}{x\left(x-4\right)}=2\Rightarrow 960=2x^2-8x\Rightarrow 2x^2-8x-960=0\\ \Rightarrow x^2-4x-480=0.\end{array}$

Recall the quadratic equation $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ where $a=1,b=-4,c=-480$ .

Therefore

  $\begin{array}{l}\frac{-\left(-4\right)\pm \sqrt{{\left(4\right)}^2+4\times 1\times 480}}{2}\Rightarrow x=\frac{4\pm \sqrt{16+1920}}{2}\Rightarrow x=\frac{4\pm \sqrt{1936}}{2}\\ \Rightarrow x=\frac{4\pm 44}{2}\Rightarrow x=\frac{46}{2},\frac{-40}{2}\Rightarrow x=23,-20.\end{array}$

Therefore total number of student went for the computer museum was $\left(x-4\right)=23-4=19$ .

Here we reject $x=-20$ as the student cannot be in negative.