<h2>
Answer:</h2>
The number of students who are going to the tournament are:
12
<h2>
Step-by-step explanation:</h2>
Let x be the number of students who were earlier going to trip.
and total cost=$ 210
Hence, the cost per person earlier was:
![\dfrac{210}{x}](https://tex.z-dn.net/?f=%5Cdfrac%7B210%7D%7Bx%7D)
Also, 3 more students joined the group.
This means that the cost per person now will be:
![\dfrac{210}{x+3}](https://tex.z-dn.net/?f=%5Cdfrac%7B210%7D%7Bx%2B3%7D)
Also, it is given that:
When 3 students who are not members of the club join the trip, the transportation cost per person drops by $5.83.
This means that:
![\dfrac{210}{x+3}=\dfrac{210}{x}-5.83](https://tex.z-dn.net/?f=%5Cdfrac%7B210%7D%7Bx%2B3%7D%3D%5Cdfrac%7B210%7D%7Bx%7D-5.83)
Hence,
![\dfrac{210}{x}-\dfrac{210}{x+3}=5.83](https://tex.z-dn.net/?f=%5Cdfrac%7B210%7D%7Bx%7D-%5Cdfrac%7B210%7D%7Bx%2B3%7D%3D5.83)
i.e.
![\dfrac{210(x+3-x)}{x(x+3)}=5.83\\\\\\i.e.\\\\\\\dfrac{210\times 3}{x(x+3)}=5.83\\\\\\i.e.\\\\\\x(x+3)=\dfrac{630}{5.83}\\\\\\x(x+3)=108.061](https://tex.z-dn.net/?f=%5Cdfrac%7B210%28x%2B3-x%29%7D%7Bx%28x%2B3%29%7D%3D5.83%5C%5C%5C%5C%5C%5Ci.e.%5C%5C%5C%5C%5C%5C%5Cdfrac%7B210%5Ctimes%203%7D%7Bx%28x%2B3%29%7D%3D5.83%5C%5C%5C%5C%5C%5Ci.e.%5C%5C%5C%5C%5C%5Cx%28x%2B3%29%3D%5Cdfrac%7B630%7D%7B5.83%7D%5C%5C%5C%5C%5C%5Cx%28x%2B3%29%3D108.061)
on simplifying the equation we get:
![583x^2+1749x-63000=0](https://tex.z-dn.net/?f=583x%5E2%2B1749x-63000%3D0)
Hence, on using the quadratic formula:
i.e. any quadratic equation of the type:
![ax^2+bx+c=0](https://tex.z-dn.net/?f=ax%5E2%2Bbx%2Bc%3D0)
has solution of the type:
![x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=x%3D%5Cdfrac%7B-b%5Cpm%20%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
Here we have:
a= 583, b=1749 and c= -63000
Hence, on putting these value in the quadratic formula we get the value of x as:
x=9.003 and x= -12.003
Since, the students will exist as a whole and positive number.
Hence, x=9
This means that the total number of students who will be going to the tournament are: x+3=9+3=12 students.