On the line provided, identify each word group by writing P for phrase or NP for not a phrase.

$\underline{\hspace{2cm}}$ since you didn't call

Bob gave four-fifths of his pencils to Barbara, then he gave two-thirds of the remaining pencils to Bonnie. If he ended up with ten pencils for himself, with how many did he start?

Considering again the argument from Question 4. Which of the following conclusions is true? a. Paul is a knight. b. Paul is not a knight. c. Paul may or may not be a knight.

Objective function:
C = 4x + 3y
Constraints:
x $\geq$ 0
2x + 3y $\geq$ 6
3x - 2y $\leq$ 9
x + 5y $\leq$ 20

The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells small radios. a. Use the formulas shown in the voice balloons to write the company's profit function, $P$, from producing and selling $x$ radios. b. Find the company's profit if 20,000 radios are produced and sold.

Perform the indicated operation. Simplify the answer when possible.

$$\frac { \sqrt { 75 } } { \sqrt { 3 } }$$

The problem involves two constraints.
Find the extreme values of the function $f(x, y, z)=x y+z^2$ on the circle in which the plane $y-x=0$ intersects the sphere $x^2+y^2+z^2=4$.

Develop a class BankAccount that supports these methods:

init(): Initializes the bank account balance to the value of the input argument, or to 0 if no input argument is given

withdraw(): Takes an amount as input and withdraws it from the balance

deposit(): Takes an amount as input and adds it to the balance

balance(): Returns the balance on the account

>>> x = BankAccount(700)
>>> x.balance()
700.00
>>> x.withdraw(70)
>>> x.balance()
630.00
>>> x.deposit(7)
>>> x.balance()
637.00

Solve for y given the set of constraints.

Given $\triangle P Q R \cong \triangle D E F, P Q=15, Q R=10, R P=8$, and $E F=6 y+4$.

Perform the indicated operation. Simplify the answer when possible.

$$\sqrt { 5 } + 4 \sqrt { 5 }$$

Use a truth table to determine whether the symbolic form of the argument is valid or invalid.

$$\begin{array}{l}{p \wedge \sim q} \\ {\frac{p}{\therefore q}}\end{array}$$

Use mathematical induction to prove that a polynomial of the form $p(t)=a_{n} t^{n}+\cdots+a_{1} t+a_{0}$ can have $n + 1$ distinct real zeros only if $a_{n}=a_{n-1}= ... = a_1 = a_0 = 0.$ [Hint: Use Rolle’s theorem.]

Write the letter for the word that best completes given sentence.
____________ The police officer's ________ statements about the confidential details of the case ended up causing _________ damage to his reputation.
a. aversive....diversionary
b. inadvertent...irreversible
c. inverse....diversified

Fill in the blank spot in the following sentence.

If citizens believe their city is ruled by gods, they might call their government a $\underline{\phantom{\text{justekst}}}$.