Elephant growth The weight W (in kilograms) of a female...

Elephant growth The weight $W$ (in kilograms) of a female African elephant at age $t$ (in years) may be approximated by $$W=2600 (1-0.51 e^{-0.075t})^{3}.$$ (a) Approximate the weight at birth. (b) Estimate the age of a female African elephant weighing 1800 kilograms by using (1) the accompanying graph and (2) the formula for $W$. (GRAPH CAN'T COPY)

Elephant growth The weight $W$ (in kilograms) of a female African elephant at age $t$ (in years) may be approximated by $$W=2600 (1-0.51 e^{-0.075t})^{3}.$$
(a) Approximate the weight at birth.
(b) Estimate the age of a female African elephant weighing 1800 kilograms by using (1) the accompanying graph and (2) the formula for $W$.
(GRAPH CAN'T COPY)

Key Concepts

Inverse Functions and Logarithms

When solving for the input that yields a particular output in functions, one often needs to use inverse operations. For exponential functions, this typically involves applying logarithms, which help 'reverse' the exponential effect and isolate the variable. This method is essential in various scientific and engineering applications where determining the time or rate from a known value is crucial.

Graphical Analysis and Estimation

Graphical analysis involves using a plot or graph of the function to visually interpret and estimate values. This method is particularly useful when an analytical solution is complex or when an intuitive understanding of how the function behaves over time is needed. It serves as a complementary approach to algebraic methods, offering a visual estimation of outputs and corresponding inputs.

Exponential Growth and Decay

Exponential functions are used to model processes that change by a constant percentage over time. In this context, the function includes an exponential decay term, illustrating how growth slows down as the subject ages. This concept is widely applicable in modeling phenomena in biology, finance, and physics, where the rate of change is discontinuous and often diminishes over time.

Function Evaluation

Function evaluation involves substituting a particular value into a function to find the corresponding output. This is a fundamental process in mathematical modeling, used to determine specific results like the initial condition or a value at a certain point in time.

Transcript

00:01 So for number 69 in section 4 .4, we're looking at a situation about a female elephant's weight, which is given by the function shown here.

00:11 W equals 2 ,600 times 1 minus 0 .51 times e, base e, to the negative 0 .075t, and then that quantity is cubed.

00:27 And there's a couple of different parts here that we're going to look at the first.

00:31 Is we're going to use this function to approximate the elephant's weight at birth.

00:38 And at birth, that would mean that the time in years is zero.

00:46 And by the way, that weight is in kilograms according to the problem.

00:51 So all we have to do for part a is just substitute zero in for the time.

01:01 And that looks like this.

01:07 So that will calculate the weight for us.

01:10 Remember that here for our base e, when we do that multiplication, that's going to give us a zero power.

01:22 And that means that e to the zero just goes to 1.

01:27 So that would be 0 .51 times 1.

01:31 And then we'll cube that value.

01:33 Whoops, let me switch back to black here.

01:35 Cube that value.

01:39 So really this is evaluating with your calculator, right? we do 1 minus 0 .051 to the third, and then multiply that by 2 ,600 and we'll approximate the weight to the nearest kilogram as 306 kilograms for that female elephant's weight.

02:04 They don't really specify what we would round to, so i'm just rounding it to the nearest kilogram.

02:11 Now, part b wants us to find the approximate age of the elephant when the elephant is at a weight of 1 ,800 kilograms.

02:31 So this time we know the weight, and we're trying to find the age in years.

02:36 And we have to do this in two ways.

02:38 One, using the graph that was given at the bottom of the problem.

02:43 And the bottom of our graph kind of starts somewhere here above like the value 500.

02:52 That's the initial one there.

02:55 And the graph curves up toward a horizontal asymptote.

03:02 That's at 2 ,600 kilograms...

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