Consider the following data table.

$$\begin{array}{|c|c|} \hline \text { Input } & \text { Output } \\ \hline 1 & 20 \\ \hline 2 & 26 \\ \hline 3 & 35 \\ \hline 4 & 41 \\ \hline 5 & 37 \\ \hline 6 & 52 \\ \hline 7 & 55 \\ \hline \end{array}$$

a. Find the balance point of the data. b. For what value b does the graph of the line y=5x+b pass through the balance point of the data?

For the following exercises, simplify each expression. $\sqrt{150}$

For the following exercises, factor the polynomials. $3 t(10 t+3)^{\frac{1}{3}}+7(10 t+3)^{\frac{4}{3}}$

For the following exercises, find the principal square root. $\sqrt[5]{-243}$

a. Find the equations of three different lines that contain the point (3, 6). b. Find the equation of the one line that contains (3, 6) and (8, -1),

A hockey team scored a total of 5 goals in the first two periods of play. If x represents the number of goals in the first period, and y represents the number of goals in the second period, make a function table of possible values for x and y.

Find the following limit or state that they do not exist. Assume a, b, c, and k are real numbers. $\lim _{x \rightarrow 3} \frac{x-3}{|x-3|}$

Find the length of the following polar curve. The complete circle r = a sin $\theta$, where a > 0.

Ultrasound, also known as sonography, is a techno logy that uses high frequency sound waves to produce images. It is used in medical applications to help doctors see inside patients' bodies. How do you think the sound waves can be used to image bones, muscles, and other internal structures?

A friend argues that the location of a petroleum deposit is a sign that marine organisms once lived there. Is your friend correct? Explain.

Draw the state graph for the machine, and compute the output sequence for the given input sequence. 0011

$$\begin{array}{lllr} \text{Present state} & \text{Next state} & \text{ } & \text{Output}\\ \text{ } & \text{Present input} & \text{ } & \text{ }\\ \text{ } & \text{0} & \text{1} & \text{ }\\ \hline \text{$s_{0}$} & \text{$s_{2}$} & \text{$s_{3}$} & \text{0}\\ \text{$s_{1}$} & \text{$s_{0}$} & \text{$s_{1}$} & \text{1}\\ \text{$s_{2}$} & \text{$s_{1}$} & \text{$s_{3}$} & \text{0}\\ \text{$s_{3}$} & \text{$s_{1}$} & \text{$s_{2}$} & \text{1}\\ \end{array}$$

Find two different trees that have the same list of nodes under a preorder traversal.

Consider a function $f\left(x_{1}, \ldots, x_{n}\right)$ of n variables. How many different partial derivatives of order r does f possess?

For the following exercises, factor by grouping. $20 w^{2}-47 w+24$