Q: A parabola $$y = ax^2$ + bx + c$$ passes through the points $$(0, 1)$$, $$(3, 4)$$, and $$(2, 9). $$Which system of equations could be used to find $$a$$, $$b$$, and $$c$$?

$$\begin{cases} 1 = c \\ 4 = 9a + 3b + c \\ 9 = 4a + 2b + c \end{cases}$$

Q: Write the augmented matrix for the system: $$\begin{cases} x + y + 2z = 4 \\ 2x + y + z = 3 \\ 3x - y + 6z = 6 \end{cases}$$

$$\begin{bmatrix} 1 & 1 & 2 & 4 \\ 2 & 1 & 1 & 3 \\ 3 & -1 & 6 & 6 \end{bmatrix}$$

Question 1

Solve the following system of equations for $$(x, y)$$:

$$\begin{cases}
2x - y = 3 \
2x + 3y = 5
\end{cases}$$

What is the solution?

Accepted Answer

$$(2, -1/2)$$

Question 2

Solve for $$t in $$the equation:

$$\log_2$ (t + 4) + \log_2$ (t + 1) = 3$$

What is the value of $$t$$?

Accepted Answer

Question 3

A city's population is modeled by $$P = 80,000 e$$^{0.035t}$$, where t = 0$ corresponds to 1995. In what year will the population double from its 1995 value?

Accepted Answer

2015

Question 4

Use properties of logarithms to write the following as a single logarithmic expression:

$$3 \ln(2x) - 2 \ln(3x)$$

Accepted Answer

$$\ln\left(\dfrac{8x^3$}{9x^2$}\right)$$

Question 5

Evaluate $$\log_{12} 47 to $$three decimal places.

Accepted Answer

1.497

Question 6

A parabola $$y = ax^2$ + bx + c$$ passes through the points $$(0, 1)$$, $$(3, 4)$$, and $$(2, 9). $$Which system of equations could be used to find $$a$$, $$b$$, and $$c$$?

Accepted Answer

$$\begin{cases} 1 = c \ 4 = 9a + 3b + c \ 9 = 4a + 2b + c \end{cases}$$

Question 7

The half-life of a radioactive isotope is 1620 years, and it decays according to $$y = 90e$$^{kt}$$. What is the value of k$?

Accepted Answer

$$k = -\dfrac{\ln 2}{1620}$$

Question 8

Using your answer for $$k in $$the previous question, how long will it take for the isotope to decay to 50 g? (Give a decimal approximation to the nearest hundredth.)

Accepted Answer

Approximately 477.12 years

Question 9

The level of sound $$\beta in $$decibels is given by $$\beta(I) = 10 \log_{10} \left(\dfrac{I}{I_0$}\right)$$, where $$I_0$$ = $$10^{-16}$$ watts/cm$$^2. $$What is the decibel level if $$I = 10$$^{-9}$$ watts/cm^2$?

Accepted Answer

70 dB

Question 10

Given the system of inequalities:

$$\begin{cases} x + 2y \geq 2 \ x - y \u003c 4 \end{cases}$$

Which of the following points is in the solution set?

Accepted Answer

$$(2, 3)$$

Question 11

A furniture company makes sofas and recliners. Each sofa requires 4 hours for assembly and 2 hours for covering. Recliners require 3 hours for assembly and 5 hours for covering. The profit for sofas is $$95 each, and the profit for recliners is 110. $$The company has 150 work-hours available for assembling and 190 for covering. Let $$x be $$the number of sofas and $$y$$ the number of recliners. What is the objective function and system of constraints?

Accepted Answer

Objective: $$P = 95x + 110y$$; Constraints: $$4x + 3y \leq 150$$, $$2x + 5y \leq 190$$, $$x \geq 0$$, $$y \geq 0$$

Question 12

Write the augmented matrix for the system:

$$\begin{cases} x + y + 2z = 4 \ 2x + y + z = 3 \ 3x - y + 6z = 6 \end{cases}$$

Accepted Answer

$$\begin{bmatrix} 1 & 1 & 2 & 4 \ 2 & 1 & 1 & 3 \ 3 & -1 & 6 & 6 \end{bmatrix}$$

Question 13

Suppose you solve the system in the previous question and find $$z = 1. $$What are the values of $$x$$ and $$y$$?

Accepted Answer

$$x = 1, y = 1$$

Question 14

Given the objective function $$P = 2x + 4y$$ and the feasible set defined by the intersection of $$x + 2y \geq 2$$ and $$x - y \u003c 4$$, which vertex of the feasible region gives the maximum value of $$P$$?

Accepted Answer

$$(2, 3)$$

Question 15

If $$\log_2$ (t + 4) + \log_2$ (t + 1) = 3$$, what is the value of $$t$$?

Accepted Answer

$$t = 3$$

Question 16

A radioactive isotope decays according to $$y = 90e$$^{kt}$$, with k$$ = -\dfrac{\ln 2}{1620}$$. How many years does it take for the isotope to decay to 25% of its original amount?

Accepted Answer

3240 years

Question 17

Which of the following is the correct application of the properties of logarithms to simplify $$2\log_3$ x - \log_3$ y$$?

Accepted Answer

$$\log_3$ \left(\dfrac{x^2$}{y}\right)$$

Question 18

If the population of a city grows according to $$P = 80,000e$$^{0.035t}$$, how long will it take for the population to triple?

Accepted Answer

Approximately 31.4 years

Question 19

Given the system $$\begin{cases} x + y + 2z = 4 \ 2x + y + z = 3 \ 3x - y + 6z = 6 \end{cases}$$, what is the value of $$z$$?

Accepted Answer

$$z = 1$$

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