Question

The spread of a fungal infection in an ant colony can be modeled by dy/dx = (y+1)/(2√t) where t is the time in hours and y is the percent of the ants infected. At t = 1 only three percent of the ants are infected.
a) Write an equation for the line tangent to the graph of y at t = 1. Use the tangent line to approximate the percentage of the colony infected at t = 1.2.
b) Find y explicitly in terms of t.

Answer 1

Answer:

Step-by-step explanation:

dy/dt=(y+1)(2√t)

separate the variables and integrating

$\int \frac{dy}{y+1} =\int\ {2\sqrt{t} } \, dt \\log(y+1)=2\frac{t^{\frac{3}{2} } }{\frac{3}{2} } +c\\y+1=e^{\frac{4}{3}t^{\frac{3}{2}+c } } =Ce^{t^{\frac{3}{2} }} \\y=Ce^{t^{\frac{3}{2} } } -1$