# 4.8 Fitting exponential models to data  (Page 7/12)

 Page 7 / 12

What might a scatterplot of data points look like if it were best described by a logarithmic model?

What does the y -intercept on the graph of a logistic equation correspond to for a population modeled by that equation?

The y -intercept on the graph of a logistic equation corresponds to the initial population for the population model.

## Graphical

For the following exercises, match the given function of best fit with the appropriate scatterplot in [link] through [link] . Answer using the letter beneath the matching graph.

$y=10.209{e}^{-0.294x}$

$y=5.598-1.912\mathrm{ln}\left(x\right)$

C

$y=2.104{\left(1.479\right)}^{x}$

$y=4.607+2.733\mathrm{ln}\left(x\right)$

B

$y=\frac{14.005}{1+2.79{e}^{-0.812x}}$

## Numeric

To the nearest whole number, what is the initial value of a population modeled by the logistic equation $\text{\hspace{0.17em}}P\left(t\right)=\frac{175}{1+6.995{e}^{-0.68t}}?\text{\hspace{0.17em}}$ What is the carrying capacity?

$P\left(0\right)=22\text{\hspace{0.17em}}$ ; 175

Rewrite the exponential model $\text{\hspace{0.17em}}A\left(t\right)=1550{\left(1.085\right)}^{x}\text{\hspace{0.17em}}$ as an equivalent model with base $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ Express the exponent to four significant digits.

A logarithmic model is given by the equation $\text{\hspace{0.17em}}h\left(p\right)=67.682-5.792\mathrm{ln}\left(p\right).\text{\hspace{0.17em}}$ To the nearest hundredth, for what value of $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ does $\text{\hspace{0.17em}}h\left(p\right)=62?$

$p\approx 2.67$

A logistic model is given by the equation $\text{\hspace{0.17em}}P\left(t\right)=\frac{90}{1+5{e}^{-0.42t}}.\text{\hspace{0.17em}}$ To the nearest hundredth, for what value of t does $\text{\hspace{0.17em}}P\left(t\right)=45?$

What is the y -intercept on the graph of the logistic model given in the previous exercise?

y -intercept: $\text{\hspace{0.17em}}\left(0,15\right)$

## Technology

For the following exercises, use this scenario: The population $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ of a koi pond over $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ months is modeled by the function $\text{\hspace{0.17em}}P\left(x\right)=\frac{68}{1+16{e}^{-0.28x}}.$

Graph the population model to show the population over a span of $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ years.

What was the initial population of koi?

$4\text{\hspace{0.17em}}$ koi

How many koi will the pond have after one and a half years?

How many months will it take before there are $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ koi in the pond?

about $\text{\hspace{0.17em}}6.8\text{\hspace{0.17em}}$ months.

Use the intersect feature to approximate the number of months it will take before the population of the pond reaches half its carrying capacity.

For the following exercises, use this scenario: The population $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ of an endangered species habitat for wolves is modeled by the function $\text{\hspace{0.17em}}P\left(x\right)=\frac{558}{1+54.8{e}^{-0.462x}},$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is given in years.

Graph the population model to show the population over a span of $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ years.

What was the initial population of wolves transported to the habitat?

$10\text{\hspace{0.17em}}$ wolves

How many wolves will the habitat have after $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ years?

How many years will it take before there are $\text{\hspace{0.17em}}100\text{\hspace{0.17em}}$ wolves in the habitat?

about 5.4 years.

Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity.

For the following exercises, refer to [link] .

 x f(x) 1 1125 2 1495 3 2310 4 3294 5 4650 6 6361

Use a graphing calculator to create a scatter diagram of the data.

Use the regression feature to find an exponential function that best fits the data in the table.

Write the exponential function as an exponential equation with base $\text{\hspace{0.17em}}e.$

$f\left(x\right)=776.682{e}^{0.3549x}$

Graph the exponential equation on the scatter diagram.

Use the intersect feature to find the value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ for which $\text{\hspace{0.17em}}f\left(x\right)=4000.$

When $\text{\hspace{0.17em}}f\left(x\right)=4000,$ $x\approx 4.6.$

For the following exercises, refer to [link] .

 x f(x) 1 555 2 383 3 307 4 210 5 158 6 122

Use a graphing calculator to create a scatter diagram of the data.

#### Questions & Answers

the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard

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