Answer:
all work shown and pictured
Put the equation in standard linear form.
![x'(t) + \dfrac{x(t)}{t + 5} = 5e^{5t}](https://tex.z-dn.net/?f=x%27%28t%29%20%2B%20%5Cdfrac%7Bx%28t%29%7D%7Bt%20%2B%205%7D%20%3D%205e%5E%7B5t%7D)
Find the integrating factor.
![\mu = \exp\left(\displaystyle \int \frac{dt}{t+5}\right) = e^{\ln|t+5|} = t+5](https://tex.z-dn.net/?f=%5Cmu%20%3D%20%5Cexp%5Cleft%28%5Cdisplaystyle%20%5Cint%20%5Cfrac%7Bdt%7D%7Bt%2B5%7D%5Cright%29%20%3D%20e%5E%7B%5Cln%7Ct%2B5%7C%7D%20%3D%20t%2B5)
Multiply both sides by
.
![(t+5) x'(t) + x(t) = 5(t+5)e^{5t}](https://tex.z-dn.net/?f=%28t%2B5%29%20x%27%28t%29%20%2B%20x%28t%29%20%3D%205%28t%2B5%29e%5E%7B5t%7D)
Now the left side the derivative of a product,
![\bigg((t+5) x(t)\bigg)' = 5(t+5)e^{5t}](https://tex.z-dn.net/?f=%5Cbigg%28%28t%2B5%29%20x%28t%29%5Cbigg%29%27%20%3D%205%28t%2B5%29e%5E%7B5t%7D)
Integrate both sides.
![(t+5) x(t) = \displaystyle 5 \int (t+5) e^{5t} \, dt](https://tex.z-dn.net/?f=%28t%2B5%29%20x%28t%29%20%3D%20%5Cdisplaystyle%205%20%5Cint%20%28t%2B5%29%20e%5E%7B5t%7D%20%5C%2C%20dt)
On the right side, integrate by parts.
![(t+5) x(t) = \dfrac15 (5t+24) e^{5t} + C](https://tex.z-dn.net/?f=%28t%2B5%29%20x%28t%29%20%3D%20%5Cdfrac15%20%285t%2B24%29%20e%5E%7B5t%7D%20%2B%20C)
Solve for
.
![\boxed{x(t) = \dfrac{5t+24}{5t+25} e^{5t} + \dfrac C{t+5}}](https://tex.z-dn.net/?f=%5Cboxed%7Bx%28t%29%20%3D%20%5Cdfrac%7B5t%2B24%7D%7B5t%2B25%7D%20e%5E%7B5t%7D%20%2B%20%5Cdfrac%20C%7Bt%2B5%7D%7D)
Answer:
1/16
Step-by-step explanation:
if half the amount of blue marbles have sparkles you need to divide the amount of blue marbles in half
so 1/2 of 1/8 is 1/16
You plug in number for number getting 194.8 which is equal to 2931