The spread of a fungal infection in an ant colony can be modeled by 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.

Hey there, Guy.

(a) y (t ) is the percent of ants that are infected. We're told that y = 3 percent of the colony is infected after t = 1 hour has passed.

The derivative dy/dt gives the slope of the tangent line to y at some time t. So the tangent line to the graph of y at the point (1, 3) has slope

and so the tangent line (call it ℓ) has equation

Using the tangent line as an approximation for y, we would estimate the infection has spread to

or about 3.4% of the colony.

(b) The given differential equation is separable as

Integrate both sides to get

Plug in the initial condition to solve for C :

Then we have the particular solution,

We can solve for y explicitly:

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