Use logarithms for these questions
After every 10 pages of painting, the ribbon in the dot matrix printer loses about 0.5% of its ink.
A) What percent of the ink is left in the ribbon after printing 100 pages and 1000 pages
B) write an equation which expresses the percent p of ink left in the ribbon as a function of the number n of pages printed.
c) Approximately how many pages can be printed before the ink content is reduced to 75% of its original content?
To solve these questions using logarithms, we need to use logarithmic functions to calculate the percent of ink left in the ribbon after printing a certain number of pages.
A) To find the percent of ink left after printing 100 pages, we can use the formula:
Percent of ink left = 100% - (0.5% * Number of pages)
Using logarithms, we can simplify this formula as follows:
Percent of ink left = 100% - (0.5% * Number of pages)
= 100 - (0.5/100 * Number of pages)
= 100 - (0.5 * Number of pages)/100
= 100 - (0.005 * Number of pages)
Now we can calculate the percent of ink left after printing 100 pages:
Percent of ink left after 100 pages = 100 - (0.005 * 100)
= 100 - 0.5
= 99.5%
Similarly, to find the percent of ink left after printing 1000 pages:
Percent of ink left after 1000 pages = 100 - (0.005 * 1000)
= 100 - 5
= 95%
Therefore, after printing 100 pages, the ribbon will have 99.5% of its ink left, and after printing 1000 pages, it will have 95% of its ink left.
B) To write the equation expressing the percent p of ink left in the ribbon as a function of the number n of pages printed, we use the formula:
p(n) = 100 - (0.005 * n)
Here, p(n) represents the percent of ink left after printing n pages.
C) To find the approximate number of pages that can be printed before the ink content is reduced to 75% of its original content, we need to solve the equation:
75 = 100 - (0.005 * n)
To solve this equation using logarithms, we can rewrite it as:
0.005 * n = 25
n = 25 / 0.005
= 5000
Therefore, approximately 5000 pages can be printed before the ink content is reduced to 75% of its original content.
A) To calculate the percent of ink left in the ribbon after printing a certain number of pages, we can use logarithms.
First, let's calculate the percent of ink left after printing 100 pages:
Each time 10 pages are printed, the ribbon loses 0.5% of its ink. So, after printing 100 pages (10 pages * 10), the ribbon loses 0.5% * 10 = 5% of its ink.
Therefore, the percent of ink left after printing 100 pages is 100% - 5% = 95%.
Now, let's calculate the percent of ink left after printing 1000 pages:
Similarly, after printing 1000 pages (10 pages * 100), the ribbon loses 0.5% * 100 = 50% of its ink.
Therefore, the percent of ink left after printing 1000 pages is 100% - 50% = 50%.
B) The equation that expresses the percent p of ink left in the ribbon as a function of the number n of pages printed can be written as:
p(n) = 100% - 0.5% * (n / 10)
In this equation, n represents the number of pages printed, and p(n) represents the percent of ink left in the ribbon.
C) To approximate the number of pages that can be printed before the ink content is reduced to 75% of its original content, we can set up the equation as:
75 = 100% - 0.5% * (n / 10)
First, let's simplify the equation:
0.5% * (n / 10) = 100% - 75%
0.5% * (n / 10) = 25%
Now, let's solve for n:
(n / 10) = 25% / 0.5%
(n / 10) = 50
Multiplying both sides of the equation by 10 gives us:
n = 50 * 10 = 500
Therefore, approximately 500 pages can be printed before the ink content is reduced to 75% of its original content.
the fraction left after n pages is
p(n) = 0.995^(n/10)
Now you can use that to answer the other two parts.