The population of a virus in a host can be modeled by the function $$ A $$ that satisfies the differential equation $$ \frac{d A}{d t}=t-\frac{1}{5} A $$, where $$ A $$ is measured in millions of virus cells and $$ t $$ is measured in days for $$ 5 \leq t

Question

The population of a virus in a host can be modeled by the function that satisfies the differential equation
, where is measured in millions of virus cells and is measured in days for . At
time days, there are 10 million cells of the virus in the host.
(a) Write an equation for the line tangent to the graph of at . Use the tangent line to approximate the
number of virus cells in the host, in millions, at time days.

Ask by Montgomery Martinez. in the United States

Jan 22,2025

UpStudy AI · Accepted Answer

To solve part (a) of the problem, we'll determine the equation of the tangent line to the function $$ A(t) $$ at $$ t = 5 $$ and use it to approximate the number of virus cells at $$ t = 7 $$ days.

### Step 1: Determine the Slope of the Tangent Line

The slope of the tangent line at $$ t = 5 $$ is given by the derivative $$ \frac{dA}{dt} $$ evaluated at $$ t = 5 $$.

Given the differential equation:
$$
\frac{dA}{dt} = t - \frac{1}{5}A
$$
At $$ t = 5 $$ days and $$ A = 10 $$ million cells:
$$
\left.\frac{dA}{dt}\right|_{t=5} = 5 - \frac{1}{5}(10) = 5 - 2 = 3
$$
So, the slope $$ m $$ of the tangent line is **3**.

### Step 2: Write the Equation of the Tangent Line

Using the point-slope form of a line:
$$
A(t) = A(t_0) + m(t - t_0)
$$
where $$ t_0 = 5 $$, $$ A(t_0) = 10 $$, and $$ m = 3 $$:
$$
A(t) = 10 + 3(t - 5)
$$
$$
A(t) = 10 + 3t - 15
$$
$$
A(t) = 3t - 5
$$

### Step 3: Approximate $$ A $$ at $$ t = 7 $$ Days

Using the tangent line equation:
$$
A(7) = 3(7) - 5 = 21 - 5 = 16
$$
So, the number of virus cells at $$ t = 7 $$ days is approximately **16 million**.

### Final Answer

**Answer:**

At $$ t = 5 $$, the tangent line has slope 3. Thus, an equation for the tangent line is

$$
A(t) = 10 + 3(t - 5)
$$

Using this line to approximate $$ A(7) $$, we find

$$
A(7) = 10 + 3(7 - 5) = 16 \text{ million cells}.
$$
At $$ t = 5 $$, the tangent line has a slope of 3. The equation of the tangent line is $$ A(t) = 10 + 3(t - 5) $$. Using this line to approximate $$ A(7) $$, the number of virus cells at $$ t = 7 $$ days is approximately **16 million**.

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