Algebra Word Problem Solutions

Solve the equation

$\frac{3}{\sqrt{2x}}$

Indicate whether the question is one of statistics. Explain. What number of songs does your MP3 player have?

The temperature dropped $2^{\circ }F$ every hour for 6 hours. What was the total number of degrees the temperature changed in the 6 hours? Explain how you determined your answer.

Determine the algebraic modeling which of the following data sets are linear and which are exponential. For the linear sets, determine the slope. For the exponential sets, determine the growth factor or the decay factor
a) $\begin{array}{|cccccccc|}\hline x& -2& -1& 0& 1& 2& 3& 4\\ y& \frac{1}{9}& \frac{1}{3}& 1& 3& 9& 27& 81\\ \hline\end{array}$

b) $\begin{array}{|cccccccc|}\hline x& -2& -1& 0& 1& 2& 3& 4\\ y& 2& 2.6& 3.2& 3.8& 4.4& 5.0& 5.6\\ \hline\end{array}$

c) $\begin{array}{|cccccccc|}\hline x& -2& -1& 0& 1& 2& 3& 4\\ y& 3.00& 5.0& 7& 9& 11& 13& 15\\ \hline\end{array}$

d) $\begin{array}{|cccccccc|}\hline x& -2& -1& 0& 1& 2& 3& 4\\ y& 5.25& 2.1& 0.84& 0.336& 0.1344& 0.5376& 0.021504\\ \hline\end{array}$

Consider the following statements.
a)The sum of a rational number and a rational number is rational. 
b)The sum of a rational number and an irrational number is irrational. 
c)The sum of an irrational number and an irrational number is irrational. 
d)The product of a rational number and a rational number is rational. 
e)The product of a rational number and an irrational number is irrational. 
f)The product of an irrational number and an irrational number is irrational.

Several models have been proposed to explain the diversification of life during geological periods. According to Benton (1997), The diversification of marine families in the past 600 million years (Myr) appears to have followed two or three logistic curves, with equilibrium levels that lasted for up to 200 Myr. In contrast, continental organisms clearly show an exponential pattern of diversification, and although it is not clear whether the empirical diversification patterns are real or are artifacts of a poor fossil record, the latter explanation seems unlikely. In this problem, we will investigate three models fordiversification. They are analogous to models for populationgrowth, however, the quantities involved have a differentinterpretation. We denote by N(t) the diversification function,which counts the number of taxa as a function of time, and by rthe intrinsic rate of diversification.
(a) (Exponential Model) This model is described by ${\displaystyle \frac{dN}{dt}=r_{e}N\left(8.86\right).}$ Solve (8.86) with the initial condition N(0) at time 0, and show that ${\displaystyle r_e}$ can be estimated from ${\displaystyle r_e=\frac{1}{t}\ln \left[\frac{N\left(t\right)}{N\left(0\right)}\right]\left(8.87\right)}$
(b) (Logistic Growth) This model is described by ${\displaystyle \frac{dN}{dt}=r_{l}N(1-\frac{N}{K})\left(8.88\right)}$ where K is the equilibrium value. Solve (8.88) with the initial condition N(0) at time 0, and show that ${\displaystyle r_l}$ can be estimated from ${\displaystyle r_l=\frac{1}{t}\ln \left[\frac{K-N\left(0\right)}{N\left(0\right)}\right]+\frac{1}{t}\ln \left[\frac{N\left(t\right)}{K-N\left(t\right)}\right]\left(8.89\right)}$ for ${\displaystyle N\left(t\right)<K.}$
(c) Assume that ${\displaystyle N\left(0\right)=1}$ and ${\displaystyle N\left(10\right)=1000.}$ Estimate ${\displaystyle r_e}$ and ${\displaystyle r_l}$ for both ${\displaystyle K=1001}$ and ${\displaystyle K=10000.}$
(d) Use your answer in (c) to explain the following quote from Stanley (1979): There must be a general tendency for calculated values of ${\displaystyle \left[r\right]}$ to represent underestimates of exponential rates,because some radiation will have followed distinctly sigmoid paths during the interval evaluated.
(e) Explain why the exponential model is a good approximation to the logistic model when ${\displaystyle \frac{N}{K}}$ is small compared with 1.

Tell whether the function represents exponential growth or exponential decay. Then graph the function. $f(x)=(0.25{)}^x$

Solve the following system of linear equations in two variables by Substitution method.
$5x-2y=-7$
$x=-2y+1$