Problem 50 - College Algebra and Calculus: An Applied Approach

Question

In a typing class, the average number \(N\) of words per minute typed after \(t\) weeks of lessons can be modeled by \(N=\frac{95}{1+8.5 e^{-0.12 t}}\) (a) Use a graphing utility to estimate the average number of words per minute typed after 10 weeks. Verify your result analytically. (b) Use a graphing utility to estimate the number of weeks required to achieve an average of 70 words per minute. (c) Does the number of words per minute have a limit as \(t\) increases without bound? Explain your answer.

Step-by-Step Solution

Answer

After calculating parts (a) and (b) and analyzing part (c), it is found that the average typing speed after ten weeks is calculated from the equation for part (a). The estimation for weeks required to attain an average speed of 70 words per minute is determined in part (b). Finally, it is confirmed in Part (c) that, as \(t\) increases, the number of words per minute typed reaches a limit of 95 words.

Step 1: Calculate Average Typing Speed after 10 Weeks

To determine this, replace \(t\) with 10 in the given equation: \[ N= \frac{95}{1+8.5 e^{-0.12 \cdot 10}} \] Solve this equation to find the value of \(N\).

Step 2: Determine Time to Achieve Speed of 70 Words per Minute

For this step, set \(N\) at 70 in the original equation and solve for \(t\). So the equation now looks like the following: \[70 = \frac{95}{1+8.5 e^{-0.12 t}}\] Solve this equation to get the value of \(t\).

Step 3: Identify if a Limit Exists

Looking at the given equation, observe that as \(t\) approaches infinity, \(e^{-0.12t}\) approaches 0. Therefore, as \(t\) becomes larger and larger, the overall expression \(\frac{95}{1+8.5 e^{-0.12 t}}\) will approach \(\frac{95}{1+0}\), or 95. This refers to a horizontal asymptote, indicating that 95 is the upper limit to the number of words per minute that can be typed overtime. Thus, as \(t\) increases without bound, \(N\) approaches 95.