Boost Your Math Modeling Skills

To write and solve the system of linear equations to find the price of 1 cup of dried fruit and 1 cup of almonds.
Given: A small bag of trail mix contains 3 cups of driedd fruit and 4 cups of almonds. A large bag contains ${\displaystyle 4\frac{1}{2}}$ cups of dried fruit and 6 cups of almonds.

The value of expression ${\displaystyle -2{\left(27\right)}^{\frac{2}{3}}}$.

The value of expression ${\displaystyle {(-3)}^{-2}}$

On a trip, the total distance (in miles) you travel in x hours is represented by the piccewise function.
$d(x)=\{\begin{array}{ll}55x,& if0\le x\le 2\\ 65x-20,& if2<x\le 5\end{array}$
How far do you travel in 4 hours?

A repair bill for your car is $553. The parts cost $265. The labor cost is $48 per hour. Write and solve an equation to find the number of hours of labor spent repairing the car.

A woodworker wants to cam at least $$ 20$ an hour making and for materials, Write and solve un inequality that represents how many hours the woodworker can spent building the cabinet.
Given: The price for raw material is $125.
The selling price of cabinet is $500

The cost (in dollars) of making b bracelets is represented by ${\displaystyle 4+5b}$. The cost (in dollars) of making b necklaces is represented by ${\displaystyle 8b+6}$. Write a polynomial that represents how much more it costs to make b necklaces than b bracelets.

You have $12 to by five goldfish for your new fish tank. Write and solve an inequality that represents the prices you can pay per fish.

A discounted amusement park ticket costs $12.95 less than the original price p. Write and solve an equation to find the original price.

To find: The dimension of the froor and the dimension of the fountain-pool area.

Solve the logarithmic equation.
The logarithm equation is ${\displaystyle \ln (5-x)-\ln (5+x)=-\ln 9}$

a) A model of the provided data in the form ${\displaystyle y=a{b}^x}$.
b) The function obtained in part (a) as an exponent function with base e.
c) To calculate: The monthly cost for January in the year 2015.

The equation ${\displaystyle y=b{x}^a}$ can be written as ${\displaystyle \log y=\log b+a\log x}$

Number of days required for a sample to decay from 10 milligrams to 1 milligram.
Given: 10 milligrams of sample decays to 5 milligrams in 23 days.

The value of ${\displaystyle f^{-1}\left(0\right)}$ using the table given.
Given: a table of values for the functions f(x) and g(x).
$\begin{array}{|cccccc|}\hline x& 0& 2& 4& 6& 8\\ f(x)& 8& 2& 0& 6& 4\\ \hline\end{array}$
$\begin{array}{|cccccc|}\hline x& 0& 2& 4& 6& 8\\ g(x)& 4& 8& 0& 2& 6\\ \hline\end{array}$

To solve: The problemby modeling and solving a system of linear equation.
The given problem is a mixer problem where ${\displaystyle 60\mathrm{\%}}$ sugar solution and ${\displaystyle 30\mathrm{\%}}$ sugar solution are mixed together to make ${\displaystyle 51\mathrm{\%}}$ sugar solution where quantity of ${\displaystyle 30\mathrm{\%}}$ sugar solution is to be calculated.

Average rate of change f(x)

The fourth term of ${\displaystyle (2x+y)5}$
A binomial is a polynomial which contains two terms. Algebraically, sum or difference of two monomials is a binomial. Binomial theorem is used to expand the binomials to any given power without direct multiplication. According to the theorem, ${\displaystyle (x+y)n=\sum k=0\cap Ckxn-kyk}$ where ${\displaystyle nCk=C(n,k)=n!(n-k)!k!}$ is the binomial coefficient in the theorem. It represents the number of ways in which k unordered outcomes can be selected from a total of n possibilities. The formula to find rth term is given as ${\displaystyle tr=nCr-1xn-r+1yr-1}$, where n is the exponent of the binomial, r is the number of the term to be found, x is the first term of the binomial and y is the second term of the binomial.

a) Show that ${\displaystyle P\left(h0\right)=K\left(vf\right)}$
b) Interpret result from part a)
Given:
A ball with mass m is dropped from an initial height of ${\displaystyle h_0}$, and lands with a final velocity of vf. The kinetic energy of the ball is ${\displaystyle K\left(v\right)=12mv2}$, where v is its velocity and the potential energy of the ball is ${\displaystyle P\left(h\right)=mgh}$, where h is its height and g is a constant.
${\displaystyle vf=2gh0}$

a) To calculate: A model of the form ${\displaystyle N\left(t\right)=a+b\ln t}$ for the provided data.
b) To calculate: The sales in a 7 weeks.
c) The validity of the statement with explanation.