The percentage of families that were married couples with...

The percentage of families that were married couples with children between 1970 and 2010 is approximately $$ P(t)=\frac{49.6}{t^{0.27}} \quad(1 \leq t \leq 5) $$ where $t$ is measured in decades, with $t=1$ corresponding to $1970 .$ a. What percentage of families were married couples with children in $1970 ?$ In 1990 ? In 2010 ? b. How fast was the percentage of families that were married couples with children changing in $1990 ?$ In $2000 ?$

The percentage of families that were married couples with children between 1970 and 2010 is approximately
$$
P(t)=\frac{49.6}{t^{0.27}} \quad(1 \leq t \leq 5)
$$
where $t$ is measured in decades, with $t=1$ corresponding to $1970 .$
a. What percentage of families were married couples with children in $1970 ?$ In 1990 ? In 2010 ?
b. How fast was the percentage of families that were married couples with children changing in $1990 ?$ In $2000 ?$

Key Concepts

Function Evaluation

This concept involves substituting specific values into a given function to determine its output. In the context of the problem, evaluating the percentage model P(t) at particular times requires plugging in the appropriate t values (in decades) to obtain the corresponding percentages.

Differentiation and Rate of Change

Differentiation is the process of finding the derivative of a function, which measures how the function's output changes with respect to changes in its input. In this problem, the derivative of P(t) with respect to time is used to calculate how fast the percentage of families is changing at given times. This connects directly to the concept of rate of change.

Power Functions

This concept pertains to functions in which the variable is raised to a constant exponent. The model provided, P(t) = 49.6/t^(0.27), is a power function. Understanding the properties of power functions, such as their growth or decay behavior and how they respond to changing variables, is essential when evaluating and differentiating such expressions.

Transcript

00:01 All right, so we've got a study that gives us the percentage of families that are married couples with children, so standard traditional family, from 1970 to 2010.

00:11 And they say that tea is measured in decades.

00:16 So first, let's go ahead and fill out this table so that we know what tea values or what year.

00:21 So two would be the 1980, three would be 1990, four would be 2000, and then, of course, five is 2010, and that works out.

00:31 Okay, so first they ask what percentage of families are married couples with children, and they want you to find the values in 1970, let's make sure, 1990 and 2010.

00:48 So we're just looking for p of 1, p of 3, and p of 5.

00:51 So p of 1 is easy, is 49 .6 over 1 to that power.

00:58 So almost half of families in 1970.

01:01 Were married couples with children, p of 3.

01:06 That actually seems low to me.

01:08 I would have expected more for 1970.

01:12 But we'll see what happens as time goes on.

01:15 So type this into your calculator and make sure you have all the parentheses correctly everywhere.

01:22 Otherwise you can get vastly different answers than what you're looking for.

01:29 Oh no, i hit clear.

01:31 Don't do that.

01:32 That will not give you the answer.

01:34 3 to the power of 0 .27.

01:37 Okay, here we go.

01:39 So as expected, it decreases a little bit to and if we head to 1990.

01:46 I'm rounding to two decimal places since it doesn't tell us what to round to.

01:51 And then p of 5 for 2010 will be 49 .6 over 5 to the 0 .27.

01:57 And then i'll just change that number in my calculator and it'll be approximately 32 .12%.

02:06 So still decreasing, but maybe not as drastically by 2010.

02:13 Okay, and then how fast was that number changing? so, okay, we'll see what we just talked about.

02:21 So before we calculate that, because we'll need a derivative for that, we'll just write this as t to the negative point...

Please give Ace some feedback