Answer:
The entire floor can contain approximately 4178 fans
Step-by-step explanation:
The first step is to calculate the area of the basketball floor.
we can do this by multiplying the length by the breadth as such 94 X 50 = 4700 square feet.
The second step is to calculate the area occupied by 8 of the fans. We can do this by multiplying 3 ft by 3ft = 9 square feet.
From this, it will be easier to estimate the area occupied by only one fan. This can be got by dividing 9 square feet by 8.
This is = 1.125 square feet.
To get the number of students it can occupy, we divide the total area of the court by the area occupied by one student.
4700/ 1.125 =4177.8
4178 fans
![f(t)=\begin{cases}\cos t&\text{for }0\le t](https://tex.z-dn.net/?f=f%28t%29%3D%5Cbegin%7Bcases%7D%5Ccos%20t%26%5Ctext%7Bfor%20%7D0%5Cle%20t%3C%5Cpi%5C%5C0%26%5Ctext%7Bfor%20%7Dt%5Cge%5Cpi%5Cend%7Bcases%7D)
Write
in terms of the step function
:
![f(t)=(u(t)-u(t-\pi))\cos t](https://tex.z-dn.net/?f=f%28t%29%3D%28u%28t%29-u%28t-%5Cpi%29%29%5Ccos%20t)
where the step function is defined by
![u(t)=\begin{cases}1&\text{for }t\ge0\\0&\text{for }t](https://tex.z-dn.net/?f=u%28t%29%3D%5Cbegin%7Bcases%7D1%26%5Ctext%7Bfor%20%7Dt%5Cge0%5C%5C0%26%5Ctext%7Bfor%20%7Dt%3C0%5Cend%7Bcases%7D)
The Laplace transform is then
![\mathcal L_s\{f(t)\}=F(s)=\displaystyle\int_0^\infty f(t)e^{-st}\,\mathrm dt=\int_0^\pi\cos t\,e^{-st}\,\mathrm dt](https://tex.z-dn.net/?f=%5Cmathcal%20L_s%5C%7Bf%28t%29%5C%7D%3DF%28s%29%3D%5Cdisplaystyle%5Cint_0%5E%5Cinfty%20f%28t%29e%5E%7B-st%7D%5C%2C%5Cmathrm%20dt%3D%5Cint_0%5E%5Cpi%5Ccos%20t%5C%2Ce%5E%7B-st%7D%5C%2C%5Cmathrm%20dt)
![\implies F(s)=\dfrac{(e^{-\pi s}+1)s}{s^2+1}](https://tex.z-dn.net/?f=%5Cimplies%20F%28s%29%3D%5Cdfrac%7B%28e%5E%7B-%5Cpi%20s%7D%2B1%29s%7D%7Bs%5E2%2B1%7D)
(if you're stuck on the integral, try integrating by parts)
We know that the bottle of soft drink has a remaining
volume which is 2 / 3 of the original.
Since Jack drank 1 / 4 of the remaining volume, hence only
3 / 4 now finally remains, the total is:
(2 / 3) * (3 / 4) = 6 / 12 = 1 / 2
<span>Hence the bottle is now one half full</span>
I might not be right but i think it’s 104