Q: Write the augmented matrix for the system: $$\begin{cases} x + 2y = 5 \\ 3x - y = 4 \end{cases}$$

$$\begin{pmatrix} 1 & 2 & 5 \\ 3 & -1 & 4 \end{pmatrix}$$

Question 1

A rational equation models the concentration of a drug in the bloodstream over time: $$C(t) = \frac{50}{t+2}. If $$the concentration must be exactly 10 mg/L, at what time $$t (in $$hours) will this occur?

Accepted Answer

$$t = 5$$

Question 2

Given the polynomial $$f(x) = 4x^5$ - 3x^3$ + 2x - 7$$, what is the degree, leading term, and leading coefficient?

Accepted Answer

Degree: $$5$$, Leading term: $$4x^5$$, Leading coefficient: $$4$$

Question 3

A polynomial function $$g(x) = (x-2)^3$(x+1)^2 is $$given. What is the behavior of the graph at $$x = -1$?

Accepted Answer

The graph touches the $$x-$$axis and turns around at $$x = -1$$

Question 4

A company models its profit with $$P(x) = x^4$ - 6x^2$ + 9. $$What is the maximum number of real zeros this polynomial can have?

Accepted Answer

4

Question 5

Which of the following is the domain of the rational function $$f(x) = \frac{2x+1}{x^2$-4}$$?

Accepted Answer

All real numbers except $$x = 2$$ and $$x = -2$$

Question 6

A population of bacteria grows according to $$P(t) = 500e$$^{0.2t}$$. What is the population after 5 hours?

Accepted Answer

$$P(5) = 500e$$^{1}$$ \approx 1359$$

Question 7

If $$f(x) = 3x + 7$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x-7}{3}$$

Question 8

A savings account earns compound interest at 6% annual rate, compounded quarterly. If you deposit $$1000, what is the balance after 2 years? (Use A = P$$\left(1 + \frac{r}{n}\$$right)^{nt}$)

Accepted Answer

$$A = 1000\left(1 + \frac{0.06}{4}\$$right)^{8}$$ \approx 1126.83$$

Question 9

Rewrite $$\log_5$ 125 as an $$exponential equation.

Accepted Answer

$$5^x = 125$$

Question 10

Evaluate $$\log_2$ 32.$$

Accepted Answer

5

Question 11

Given $$\log_3$ 81 = x$$, what is the value of $$x$$?

Accepted Answer

4

Question 12

Rewrite $$\log_6$ 18$$ using the change of base formula.

Accepted Answer

$$\frac{\log_{10} 18}{\log_{10} 6}$$

Question 13

If $$\log_2$ x = 3$$, what is $$x$$?

Accepted Answer

$$x = 8$$

Question 14

Solve the exponential equation $$2^{x} = 16$.

Accepted Answer

$$x = 4$$

Question 15

A radioactive substance decays according to $$N(t) = 1000e$$^{-0.05t}$$. How much remains after 10 years?

Accepted Answer

$$N(10) = 1000e$$^{-0.5}$$ \approx 606.53$$

Question 16

Solve the system of equations: $$\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$$

Accepted Answer

$$x = 2, y = 1$$

Question 17

What is the order of the matrix $$\begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{pmatrix}$$?

Accepted Answer

2 rows by 3 columns

Question 18

Write the augmented matrix for the system: $$\begin{cases} x + 2y = 5 \ 3x - y = 4 \end{cases}$$

Accepted Answer

$$\begin{pmatrix} 1 & 2 & 5 \ 3 & -1 & 4 \end{pmatrix}$$

Question 19

A quadratic equation models the height of a ball: $$h(t) = -16t^2$ + 32t + 5. $$What is the maximum height reached by the ball?

Accepted Answer

Maximum at $$t = 1$$, $$h(1) = 21$$ feet

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