Boost Your Math Modeling Skills

A number of common substances are Tar Sand “Silly Putty”, Jello Modeling clay, Tooth paste, Wax, and Shaving cream. Some of these materials exhibit characteristics of both solid and fluid behavior under different conditions. Explain and give examples.

Modeling Luxury Home Sales
The following quadratic function in general form, ${\displaystyle S\left(t\right)=5.8t^2-81.2t+1200}$ models the number of luxury home sales, S(t), in a major Canadian urban area, according to statistical data gathered over a 12 year period. Luxury home sales are defined in this market as sales of properties worth over $3 Million (inflation adjusted). In this case, ${\displaystyle t=0}$ represents 2000 and ${\displaystyle t=11}$ represents 2011. Use a calculator to find the year when the smallest number of luxury home sales occurred. Without sketching the function, interpret the meaning of this function, on the given practical domain, in one well-expressed sentence.

Whether the given number represents a probability.
Given: 2.5

The function represents the height y (in feet) of a grasshopper jumping straight up from the ground f seconds after the start of the jump. After how many seconds is the grasshopper 1 foot oft the ground?
Given: Function ${\displaystyle y=-16t^2+8t}$ represents the height (in feet) of grasshopper at t seconds.

The function ${\displaystyle C\left(x\right)=17.5x-10}$ represents the cost (in dollars) of buying x tickets to the orchestra with a $10 coupon.
a. How much does it cost to buy five tickets?
b. How many tickets can you buy with $130?

The inverse, domain and range of inverse and verify that they are inverses of each other.
Given: The function ${\displaystyle f\left(x\right)=2x^3}$

Whether the given data is linear or exponential and find the corresponding model.
Given: The table
$\begin{array}{|cccccc|}\hline x& 0& 1& 2& 3& 4\\ y& 2& 0.8& -0.4& -1.6& -2.8\\ \hline\end{array}$

The value of m and b for the linear function ${\displaystyle f\left(x\right)=-8x}$

a) Find the mass density of a proton, modeling it as a solid sphere of radius ${\displaystyle 1.00\times 10-15m.}$
b) What If ? Consider a classical model of an electron as a uniform solid sphere with the same density as the proton. Find its radius.
c) Imagine that this electron possesses spin angular momentum ${\displaystyle lw=\frac{h}{2}}$ because of classical rotation about the z axis. Determine the speed of a point on the equator of the electron.
d) State how this speed compares with the speed of light.

The interval notation of the given inequality
GIven: ${\displaystyle \frac{2x-5}{2}<\frac{5x+1}{5}}$

The normal body temperature of a camel is ${\displaystyle {37}^{\circ }C}$. This temperature varies by up ${\displaystyle 3^{\circ }}$ throughout the day. Write and solve an absolute value inequality that represents the range of normal body temperatures (in degrees Celsius) of a camel throughout the day.
Given information:
Normal temperature of camel ${\displaystyle ={37}^{\circ }C}$
This temperature can vary by up to ${\displaystyle 3^{\circ }C}$

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? summation n=1 to infinity ${\displaystyle {(-1)}^n+\frac{1}{n^6}}$ (|error|<0.00005)

A stained glass artist has a fixed cost of $150. It costs the artist $15 to procedure each piece, but each piece sells for $35. The equation ${\displaystyle C=150+15n}$ represents the total cost, C, for producing n pieces. The total revenue for n pieces is determined by the equation ${\displaystyle R=35n}$. What constaint is necessary to include when modeling this situation?
- n must be greater than or equal to 0, since the number of items created cannot be negative.
- R must be greater than or equal to 35, since the number of items will always be positive.
- R-C must be less than or equal to 0, since the artist wants to make money.
- C must be 150, since there is a fixed cost to produce each piece.

Is a linear, power, or exponential function appropriate for modeling these data?
x136101416 y7.224.612.380.990.410.26

Weather forecasting and severe weather warning systems rely on data collection, data analysis, and computer modeling. They also rely on fast communication systems. Explain why communication is an important part of forecasting and warning systems.

Data points (x, y) are given. Is a linear, power, or exponential function appropriate for modeling these data?
x 2 4 6 8 10 12 y 0.08 0.12 0.18 0.26 0.35 0.53

a) Whether S is a function or not.
b) Domain and Range of S

The value of expression ${\displaystyle {\log }_{8}64}$

To find the measure of diversity H for the following proportions of each:
hemlock, 0.521; beech, 0.324; birch, 0.081; maple, 0.074.