Problem 104 - College Algebra Essentials

Question

The formula $$t=\frac{1}{c}[\ln A-\ln (A-N)]$$ describes the time, \(t,\) in weeks, that it takes to achieve mastery of a portion of a task, where \(A\) is the maximum learning possible, \(N\) is the portion of the learning that is to be achieved, and \(c\) is a constant used to measure an individual's learning style. a. Express the formula so that the expression in brackets is written as a single logarithm. b. The formula is also used to determine how long it will take chimpanzees and apes to master a task. For example, a typical chimpanzee learning sign language can master a maximum of 65 signs. Use the form of the formula from part (a) to answer this question: How many weeks will it take a chimpanzee to master 30 signs if \(c\) for that chimp is \(0.03 ?\)

Step-by-Step Solution

Answer

The chimpanzee will master 30 signs in approximately 21 weeks.

Step 1: Express the formula as a single logarithm

We start by using properties of logarithms to simplify the expression inside the square brackets. The given expression is \(ln(A) - ln(A-N)\). Based on the logarithmic rule which states that the difference of two logarithms can be expressed as the logarithm of a fraction, we can write this as \(ln(\frac{A}{A-N})\). Therefore, the time formula becomes \(t=\frac{1}{c}ln(\frac{A}{A-N})\).

Step 2: Substitute the given values into the formula

For the given case, the maximum number of signs a chimp can learn \(A\) is 65, the goal number of signs to be learned \(N\) is 30, and the constant \(c\) is 0.03. Plugging these values into the equation: \(t=\frac{1}{0.03}ln(\frac{65}{65-30})\).

Step 3: Calculate the value

By simplifying the equation, first subtract 30 from 65 in the brackets, which leads to \(t=\frac{1}{0.03}ln(\frac{65}{35})\). Next, calculate the fraction inside the logarithm: \(ln(\frac{65}{35})=0.616186139\), then substitute this back into the equation to obtain \(t=\frac{1}{0.03}(0.616186139)\). After multiplying and rounding to nearest week, you get \(t \approx 21\) weeks.