Question

1. reading mario has a 500 - page novel which he is required to read over summer break for his upcoming language arts class.

a. if mario reads 24 pages each day, write and graph a function to represent the number of pages that mario has left to read ( p ) after ( d ) days.
b. estimate the zero of the function by using the graph. justify your response.
c. find the zero algebraically. interpret its meaning in the context of the situation.

Explanation:

Step1: Write the function

The total number of pages is \(500\). Mario reads \(24\) pages per day. So the number of pages read after \(d\) days is \(24d\). The number of pages left \(p\) is given by the linear function \(p(d)=500 - 24d\).

Step2: Graph the function

The function \(p(d)=500 - 24d\) is a linear function. The \(y\) - intercept (when \(d = 0\)) is \(p(0)=500\). To find another point, when \(d = 10\), \(p(10)=500-24\times10=500 - 240=260\). Plot the points \((0,500)\) and \((10,260)\) and draw a straight line through them (since \(d\geq0\) and \(p\geq0\), we consider the first - quadrant part of the line).

Step3: Estimate the zero from the graph (for part b)

The zero of the function is the value of \(d\) when \(p(d)=0\). Looking at the graph (by extending the line if needed), we can estimate. Since \(p(d)=0\) means \(500-24d = 0\) or \(24d=500\), \(d=\frac{500}{24}\approx20.83\). From the graph, we can estimate \(d\approx21\) (by seeing where the line crosses the \(d\) - axis).

Step4: Find the zero algebraically (for part c)

Set \(p(d)=0\).

$$
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Answer:

a. The function is \(p(d)=500 - 24d\).
b. Estimated zero: \(d\approx21\) (by looking at the \(d\) - value where the graph of \(p(d)=500 - 24d\) crosses the \(d\) - axis).
c. Algebraic zero: \(d = \frac{125}{6}\approx20.83\). This means that after approximately \(20.83\) days, Mario will have read all \(500\) pages of the novel.