Question

1. The number of people living with HIV infections increased exponentially worldwide from 1990 to 2005. The formula for the number of HIV infections, H, in millions, as a function of years, t, since 1990 is given by H = 2.3e ^(0.1343t) A) How many people were living with HIV infection in 1990? Show how you calculated this value.B) Use functional notation to express the number of people infected in the year 2003, then calculate thatC) Explain the meaning of H (5) and calculate H (5)

Answer 1

`\begin{gathered} a)H=2.3e^((0.1343\cdot0)),\: 2.3millions\: HIV-positive \\ b)H(13)=2.3^((0.1343\cdot13))=13.181\: millions\: of\: HIV-positive \\ c)H(5)\: is\: the\: number\: of\: HIV-positive\: 5years\: later \end{gathered}`

1) In this problem, let's work with an exponential model.

a) To find out how many people were living with HIV infection we're going to consider 1990 as the first year. The year "0"

So we can write out

`\begin{gathered} H(t)=2.3e^(0.1343t) \\ H=2.3e^(0.1343*0) \\ H=2.3\cdot1 \\ H=2.3 \end{gathered}`

Note that we plugged into t, t=0 then we know that according to that exponential model there were 2.3 million people HIV positive in 1990.

b) Now we can evaluate the function, 13 years later in 2003, so we'll plug into that t=13:

`\begin{gathered} H(t)=2.3e^(0.1343t) \\ H(13)=2.3e^(0.1343\cdot13) \\ H(13)=13.18143 \end{gathered}`

So 13 years later there were more than 13 million people infected

c) In this case, the meaning of H(5) is the number of HIV positive 5 years later since t is given in years. In other words, H(5) is the number of HIV positive in 1995.