# Consider that algebraic modeling Identify the decay factor, if any, for each function. If there is no decay factor, type N. 1) displaystyle{y}={0.8}^{x} 2) displaystyle{y}=-{3}{x}-{8} 3) displaystyle g{{left({x}right)}}={left(frac{1}{{2}}right)}^{x}

Question
Modeling
Consider that algebraic modeling Identify the decay factor, if any, for each function. If there is no decay factor, type N.
1) $$\displaystyle{y}={0.8}^{x}$$
2) $$\displaystyle{y}=-{3}{x}-{8}$$
3) $$\displaystyle g{{\left({x}\right)}}={\left(\frac{1}{{2}}\right)}^{x}$$

2020-11-06
Given, 1) $$\displaystyle{y}={0.8}^{x}$$
2) $$\displaystyle{y}=-{3}{x}-{8}$$
3) $$\displaystyle g{{\left({x}\right)}}={\left(\frac{1}{{2}}\right)}^{x}$$
we have to find decay factor
Here 1) $$\displaystyle{y}={0.8}^{x}$$ which is exponential function of the form of
$$\displaystyle{y}={a}{\left({b}\right)}^{x}\text{where}\ {a}={1},{b}={0.8}$$
Hence, the decay factor is: 0.8
2) $$\displaystyle{y}=-{3}{x}-{8},$$ Decay factor is N
3) $$\displaystyle g{{\left({x}\right)}}={\left(\frac{1}{{2}}\right)}{x},$$
Decay factor is $$\displaystyle\frac{1}{{2}}$$

### Relevant Questions

Consider that algebraic modeling For the function $$\displaystyle f{{\left({x}\right)}}={34}{\left({1.024}\right)}^{x}$$
1) The function is an increasing exponential function because it is the form $$\displaystyle{y}={a}{b}^{x}$$ and ?
2) The growth rate is ?
3) Thrawth factor is ?
Determine the algebraic modeling a.
One type of Iodine disintegrates continuously at a constant rate of $$\displaystyle{8.6}\%$$ per day.
Suppose the original amount, $$\displaystyle{P}_{{0}}$$, is 10 grams, and let t be measured in days.
Because the Iodine is decaying continuously at a constant rate, we use the model $$\displaystyle{P}={P}_{{0}}{e}^{k}{t}$$ for the decay equation, where k is the rate of continuous decay.
Using the given information, write the decay equation for this type of Iodine.
b.
Use your equation to determine the half-life ofthis type of Fodine, That is, find ‘out how many days it will take for half of the original amount to be left. Show an algebraic solution using logs.
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
Determine the algebraic modeling The personnel costs in the city of Greenberg were \$9,500,000 in 2009. In the recent past the personnel costs have increased at the rate of $$4.2\%$$ annually.
To model this, the city manager uses the function $$\displaystyle{C}{\left({t}\right)}={9.5}{\left({1}{042}\right)}^{t}\text{where}\ {C}{\left({t}\right)}$$ is the annual personnel costs,in millions of dollars, t years past 2009
1) Write the equation that can be solved to find in what year the personnel costs will be double the 2009 personnel costs.
2) Then solve the equation numerically (use the Table feature of your calculator) or graphically and determine the year.
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
Consider that math modeling following initial valu problem
$$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{t}\right.}}={3}-{2}{t}-{0.5}{y},{y}{\left({0}\right)}={1}$$
We would liketo find an approximation solution with the step siza $$h = 0.05$$
What is the approximation of $$y(0.1)?$$
Determine the algebraic modeling
Solve for x: $$\displaystyle{32}{\left({1.05}\right)}^{x}={90}$$
For digits before decimals point, multiply each digit with the positive powers of ten where power is equal to the position of digit counted from left to right starting from 0.
For digits after decimals point, multiply each digit with the negative powers of ten where power is equal to the position of digit counted from right to left starting from 1.
1) $$10^{0}=1$$
2) $$10^{1}=10$$
3) $$10^{2}=100$$
4) $$10^{3}=1000$$
5) $$10^{4}=10000$$
And so on...
6) $$10^{-1}=0.1$$
7) $$10^{-2}=0.01$$
8) $$10^{-3}=0.001$$
9) $$10^{-4}=0.0001$$
$$\displaystyle{g{{\left({t}\right)}}}={2}{\left({\frac{{{5}}}{{{4}}}}\right)}^{{t}}$$
Consider a capital budgeting problem with six projects represented by $$0-1\ \text{variables}\ x1,\ x2,\ x3,\ x4,\ x5,\ \text{and}\ x6.$$
b. In which situation the constraint "$$x3\ -\ x5 = 0$$" is used, explain clearly:
$$\displaystyle{x}{4}\le{x}{1}$$
$$\displaystyle{x}{4}\le{x}{3}$$
$$\displaystyle{x}{4}\ge{x}{1}+{x}{3}-{1}$$