Question

the function p(x) = 80 - 25 log₂x models the percentage, p(x), of students who could recall the important features of a classroom lecture as a function of time, where x represents the number of days that have elapsed since the lecture was given. the figure shows the graph of the function. after how many days do only a quarter of the students recall the important features of the classroom lecture? (let p(x) = 25 and solve for x.) locate the point on the graph that conveys this information. after how many days do only a quarter of the students recall the important features of the classroom lecture? (round to one decimal place as needed.)

Explanation:

Step1: Set up the equation

We know that a quarter of the students means \( P(x) = 25 \) (since the total percentage is related to the function, and a quarter here is 25% as per the problem's context of \( P(x) = 25 \) for a quarter). The function is \( P(x)=80 - 25\log_{2}x \). So we set up the equation:
\( 25=80 - 25\log_{2}x \)

Step2: Isolate the logarithmic term

Subtract 80 from both sides:
\( 25-80=- 25\log_{2}x \)
\( - 55=- 25\log_{2}x \)
Divide both sides by - 25:
\( \log_{2}x=\frac{-55}{-25}=\frac{11}{5} = 2.2 \)

Step3: Solve for x

Using the definition of logarithms, if \( \log_{b}a = c \), then \( a = b^{c} \). Here \( b = 2 \), \( c = 2.2 \), so:
\( x = 2^{2.2} \)
Calculate \( 2^{2.2}\approx2^{2 + 0.2}=2^{2}\times2^{0.2}=4\times2^{0.2} \). We know that \( 2^{0.2}\approx1.1487 \), so \( x\approx4\times1.1487 = 4.5948\approx4.6 \) (rounded to one decimal place)

Answer:

\( 4.6 \)