Question

The number N of college graduates and thousands after T years can be modeled by n=46log(T+3) what T equal to zero represent 1985. a) how many college graduates were there in 2010? b) how long until there are 123,000 graduates when will this occur?

Answer 1

The number N of college graduates in thousands after T years can be modeled by the equation n=46log(T+3), where T=0 represents 1985 1.

a) To find the number of college graduates in 2010, we need to substitute T=25 into the equation since 2010 is 25 years after 1985. Therefore, n=46log(25+3)=46log(28)≈114.8 thousand college graduates 1.

b) To find out how long it will take for there to be 123,000 college graduates, we need to solve the equation n=46log(T+3) for T. First, we divide both sides by 46 to get log(T+3)=n/46. Then, we raise both sides as a power of e to get T+3=e^(n/46). Finally, we subtract 3 from both sides to get T=e^(n/46)-3. Substituting n=123 into this equation, we get T=e^(123/46)-3≈44.2 years after 1985, which is around 2029

Answer 2

T will be a very large number, which indicates that it will take an incredibly long time (practically impossible) to reach 123,000 graduates based on this model. This is because logarithmic growth tends to slow down over time, making it difficult to reach extremely high values within a reasonable time frame.

Explanation:

Let's solve the problems step by step:

a) To find out how many college graduates there were in 2010, we can plug in T = 2010 - 1985 = 25 into the equation:

N = 46 * log(T + 3)

N = 46 * log(25 + 3)

N = 46 * log(28)

Now, calculate the value of N using the base-10 logarithm:

N ≈ 46 * 1.4472 (rounded to four decimal places)

N ≈ 66.4906

So, there were approximately 66,491 college graduates in 2010.

b) To find out how long it will take until there are 123,000 graduates, we can rearrange the equation and solve for T:

N = 46 * log(T + 3)

123,000 = 46 * log(T + 3)

Now, divide both sides by 46:

log(T + 3) = 123,000 / 46

log(T + 3) ≈ 2673.9130

Now, we need to get rid of the logarithm by taking the inverse logarithm (base 10) on both sides:

T + 3 ≈ 10^2673.9130

T + 3 ≈ a very large number (impractical to write out)

Now, subtract 3 from both sides to isolate T:

T ≈ a very large number - 3