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
I have no idea how to solve this question. I need help.
see related question below
For the related question below, you did part B instead of A. I need help for part A.
To solve this question, we can use logarithms and the concept of percent decrease.
Let's start with finding the percent of ink left in the ribbon after printing 100 pages.
Step 1: Find the percent decrease in ink for every 10 pages.
The ribbon loses 0.5% of its ink after every 10 pages. Therefore, the percent decrease in ink per 10 pages is 0.5%.
Step 2: Determine the number of 10-page increments in 100 pages.
Since each increment is 10 pages, the number of increments in 100 pages is 100 ÷ 10 = 10 increments.
Step 3: Calculate the overall percent decrease in ink.
Using the formula for compound interest, we can calculate the overall percent decrease in ink:
(1 - 0.5%)^10
Now, let's solve this using logarithms:
Step 4: Take the logarithm to simplify the calculation.
log[(1 - 0.5%)^10]
Step 5: Use the logarithm property to simplify the expression.
10 × log(1 - 0.005)
Step 6: Evaluate the logarithm using a calculator.
10 × log(0.995) ≈ -0.502
Step 7: Convert the result back to a percentage.
The overall percent decrease in ink is approximately -0.502%.
Step 8: Determine the percent of ink remaining.
To find the percent of ink remaining, subtract the percent decrease from 100%:
100% - (-0.502%) ≈ 100.502%
Therefore, after printing 100 pages, there will be approximately 100.502% of the ink left in the ribbon.
Now, let's repeat the steps to find the percent of ink left after printing 1000 pages:
Step 2: Determine the number of 10-page increments in 1000 pages.
1000 ÷ 10 = 100 increments.
Step 3: Calculate the overall percent decrease in ink.
Using the formula for compound interest, we can calculate the overall percent decrease in ink:
(1 - 0.5%)^100
Step 4: Take the logarithm to simplify the calculation.
log[(1 - 0.5%)^100]
Step 5: Use the logarithm property to simplify the expression.
100 × log(1 - 0.005)
Step 6: Evaluate the logarithm using a calculator.
100 × log(0.995) ≈ -5.024
Step 7: Convert the result back to a percentage.
The overall percent decrease in ink is approximately -5.024%.
Step 8: Determine the percent of ink remaining.
To find the percent of ink remaining, subtract the percent decrease from 100%:
100% - (-5.024%) ≈ 105.024%
Therefore, after printing 1000 pages, there will be approximately 105.024% of the ink left in the ribbon.
In both cases, the answer is greater than 100% because the percent decrease is less than 1%.