Suppose the population $P$ of a certain species of fish depends on the number $x$ (in hundreds) of a smaller fish that serves as its food supply, so that

$$P(x)=2 x^{2}+1.$$

Suppose, also, that the number of the smaller species of fish depends on the amount a (in appropriate units) of its food supply, a kind of plankton. Specifically, $x=f(a)=3 a+2$. A biologist wants to find the relationship between the population $P$ of the large fish and the amount a of plankton available, that is,$P[f(a)]$. What is the relationship?

How many miles does a car traveling at a mph cover in one hour?

Use the duck principle to show that each equation is true. a. $\sqrt[3]{8} = 2$ b. $\sqrt{5} \cdot \sqrt{2} = \sqrt{10}$ c. $\sqrt[4]{2} \cdot \sqrt[4]{8} = 2$ d. $\sqrt[3]{4} \cdot \sqrt[3]{54} = 6$

Solve each proportion for x.

$$\frac { a + b } { c } = \frac { x } { d }$$

Which gases are In plasma?

Solve each logarithmic equation. $\log _ { 6 } ( x + 2 ) + \log _ { 6 } ( x - 3 ) = 1$

What is the best way to improve the paragraph's organization: A) Move sentence 10 to after sentence 11; B) Move sentence 9 to after sentence 12; C) Move sentence 1 to after sentence 2; D) Make no change.

Pat does a lot of travelling in her car, and she keeps records on how often she fills her car with petrol. The table alongside shows the frequencies of the number of days between refills. Estimate the likelihood that:

$$\begin{matrix} \text{Days between refills} & \text{Frequency}\\ \text{1} & \text{37}\\ \text{2} & \text{81}\\ \text{3} & \text{48}\\ \text{4} & \text{17}\\ \text{5} & \text{6}\\ \text{6} & \text{1}\\ \end{matrix}$$

there is at least a four day gap between refills.

Find the LCM of the pair of numbers. 6 and 10 ________

The cost of renting a mountain bike is $25 for the first 3 hours and$5 for each additional hour. Sketch a graph of the cost of renting a mountain bike for 0 to 8 hours. Then write a piecewise function for the graph.

Identify the block, period, group, group name (where appropriate), element name, element type, and relative reactivity for the elements with the following electron configurations. a.

$$[ \mathrm { Ne } ] 3 s ^ { 2 } 3 p ^ { 1 }$$

, b.

$$[ \mathrm { Ar } ] 3 d ^ { 10 } 4 s ^ { 2 } 4 p ^ { 6 }$$

, c.

$$[ \mathrm { Kr } ] 4 d ^ { 10 } 5 s ^ { 1 }$$

, d.

$$[ \mathrm { Xe } ] 4 ^ { 1 } 5 d ^ { 1 } 6 s ^ { 2 }$$

Write each amount as a percent. a. 17/100 b. 23 out of 100

Radar is characterized by a pulse width and pulse repetition period. (a) Explain what these words mean and how you can use them to estimate the range of distances that radar can measure. (b) If you wish to be able to detect objects at a close distance, what parameter of the radar should you change and how? If you wish to be able to detect objects at a long distance, what parameter of the radar should you change and how?

Suppose that A is a $4\times 4$ matrix in the block form, $$ \begin{equation*}A=\begin{bmatrix}B&&C\\text{\O}&&D\end{bmatrix}\end{equation*} $$ where the blocks are $2\times 2$ matrices. (a) Use a random matrix for A to illustrate that $$ \begin{equation*}A^2=\begin{bmatrix}B^2&&*\\text{\O}&&D^2\end{bmatrix}\end{equation*} $$ where * represents some $2\times2$ matrix. (b) Use a random matrix for A to illustrate that $$ \begin{equation*}A^3=\begin{bmatrix}B^3&&*\\text{\O}&&D^3\end{bmatrix}^k\end{equation*} $$ where * represents some $2\times2$ matrix. (c) Make a conjecture about the block form of $A^k$, where k is a positive integer. (d) Prove your conjecture for k=3.