Precalculus Practice Test 3: Step-by-Step Study Guidance

Q1. Solve the following system of equations algebraically and write your answer as an ordered pair:

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

Background

Topic: Systems of Linear Equations

This question tests your ability to solve a system of two linear equations in two variables using algebraic methods such as substitution or elimination.

Key Terms and Formulas

• System of equations: A set of two or more equations with the same variables.

• Elimination method: Add or subtract equations to eliminate one variable.

• Substitution method: Solve one equation for one variable and substitute into the other.

Step-by-Step Guidance

1. Choose a method (elimination or substitution). For elimination, try to align coefficients for one variable.

2. Multiply one or both equations if necessary so that the coefficients of either $x$ or $y$ are opposites.

3. Add or subtract the equations to eliminate one variable, then solve for the remaining variable.

4. Substitute the value you found back into one of the original equations to solve for the other variable.

5. Write your solution as an ordered pair $(x, y)$.

Try solving on your own before revealing the answer!

Final Answer: (2, 0)

Solving the system, we find $x = 2$ and $y = 0$. The ordered pair is $(2, 0)$.

Q2. Solve for $t$ algebraically, showing all of your work:

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

Background

Topic: Logarithmic Equations

This question tests your ability to solve equations involving logarithms, using properties of logarithms and exponentiation.

Key Terms and Formulas

• Product property: $\log_b A + \log_b B = \log_b (AB)$

• Definition of logarithm: $\log_b x = y \iff b^y = x$

Step-by-Step Guidance

1. Combine the two logarithms using the product property: $\log_2((t+3)(t+4)) = 1$.

2. Rewrite the equation in exponential form: $2^1 = (t+3)(t+4)$.

3. Expand the right side and set the equation equal to 2.

4. Rearrange the equation to standard quadratic form and prepare to solve for $t$.

Try solving on your own before revealing the answer!

Final Answer: $t = -1$

Solving the quadratic equation, we find $t = -1$ (the other solution is extraneous).

Q3. The population of a city is given by $P = 80,000 e^{0.035t}$, where $t = 0$ corresponds to the year 1995. Determine the year during which the population will be double what it was in 1995.

Background

Topic: Exponential Growth

This question tests your understanding of exponential growth models and how to solve for time when the population doubles.

Key Terms and Formulas

• Exponential growth model: $P = P_0 e^{kt}$

• Doubling time: Set $P = 2P_0$ and solve for $t$.

• Natural logarithm: $\ln x$ is the logarithm base $e$.

Step-by-Step Guidance

1. Set up the equation for doubling: $2 \times 80,000 = 80,000 e^{0.035t}$.

2. Divide both sides by 80,000 to isolate the exponential term.

3. Take the natural logarithm of both sides to solve for $t$.

4. Rearrange the equation to express $t$ in terms of logarithms.

Try solving on your own before revealing the answer!

Final Answer: Year 2015

Solving, $t \approx 20$, so the year is $1995 + 20 = 2015$.

Q4a. Use properties of logarithms to write the following as a single log expression:

$2 \ln 3x - 6 \ln 2x$

Background

Topic: Logarithmic Properties

This question tests your ability to use properties of logarithms (product, quotient, and power rules) to combine multiple logarithmic terms into a single expression.

Key Terms and Formulas

• Power rule: $a \ln b = \ln b^a$

• Difference rule: $\ln A - \ln B = \ln \left(\frac{A}{B}\right)$

Step-by-Step Guidance

1. Apply the power rule to both terms: $2 \ln 3x = \ln (3x)^2$, $6 \ln 2x = \ln (2x)^6$.

2. Rewrite the expression as $\ln (3x)^2 - \ln (2x)^6$.

3. Use the difference rule to combine into a single logarithm.

Try solving on your own before revealing the answer!

Final Answer: $\ln \left(\frac{9x^2}{64x^6}\right)$

Combining, we get $\ln \left(\frac{(3x)^2}{(2x)^6}\right) = \ln \left(\frac{9x^2}{64x^6}\right)$.

Q4b. Evaluate $\log_{12} 47$ as a decimal correct to 3 decimal places.

Background

Topic: Logarithms and Change of Base Formula

This question tests your ability to evaluate logarithms with bases other than 10 or $e$ using the change of base formula.

Key Terms and Formulas

• Change of base formula: $\log_b a = \frac{\log_c a}{\log_c b}$ (commonly $c=10$ or $c=e$)

Step-by-Step Guidance

1. Apply the change of base formula: $\log_{12} 47 = \frac{\log_{10} 47}{\log_{10} 12}$.

2. Use a calculator to find $\log_{10} 47$ and $\log_{10} 12$ to several decimal places.

3. Divide the two results to get the decimal value.

Try solving on your own before revealing the answer!

Final Answer: 1.505

$\log_{12} 47 \approx 1.505$ (rounded to three decimal places).

Q5a. A parabola $y = ax^2 + bx + c$ passes through the points (0, 3), (1, 4), and (2, 9). Determine a system of equations you could use to find the equation of the parabola.

Background

Topic: Systems of Equations for Quadratic Functions

This question tests your ability to set up a system of equations using given points on a parabola to solve for the coefficients $a$, $b$, and $c$.

Key Terms and Formulas

• Quadratic function: $y = ax^2 + bx + c$

• Substitute each point $(x, y)$ into the equation to get three equations.

Step-by-Step Guidance

1. Substitute $(0, 3)$ into $y = ax^2 + bx + c$ to get the first equation.

2. Substitute $(1, 4)$ into the equation for the second equation.

3. Substitute $(2, 9)$ into the equation for the third equation.

4. Write out the resulting system of three equations in $a$, $b$, and $c$.

Try solving on your own before revealing the answer!

Final Answer:

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

Q5b. Solve the system algebraically and give the equation of the parabola.

Background

Topic: Solving Systems of Linear Equations

This question tests your ability to solve a system of three equations in three variables to find the coefficients of a quadratic function.

Key Terms and Formulas

• Use substitution or elimination to solve for $a$, $b$, and $c$.

Step-by-Step Guidance

1. From the previous part, use $c = 3$ to substitute into the other two equations.

2. Solve the resulting two equations in $a$ and $b$.

3. Find the values of $a$ and $b$ step by step.

4. Write the final equation $y = ax^2 + bx + c$ with the values you found.

Try solving on your own before revealing the answer!

Final Answer: $y = 2x^2 - x + 3$

Solving, $a = 2$, $b = -1$, $c = 3$. The equation is $y = 2x^2 - x + 3$.

Q6a. The half-life for a radioactive isotope is 1620 years, and it decays according to the model $y = 90e^{kt}$. Find the value of $k$. Show your algebraic work and give an exact answer.

Background

Topic: Exponential Decay and Half-Life

This question tests your understanding of exponential decay and how to relate half-life to the decay constant $k$.

Key Terms and Formulas

• Exponential decay model: $y = y_0 e^{kt}$

• Half-life: The time $T$ such that $y = \frac{1}{2}y_0$

Step-by-Step Guidance

1. Set $y = \frac{1}{2} \times 90$ and $t = 1620$ in the decay model.

2. Write the equation: $45 = 90e^{k \cdot 1620}$.

3. Divide both sides by 90 to isolate the exponential term.

4. Take the natural logarithm of both sides to solve for $k$.

Try solving on your own before revealing the answer!

Final Answer: $k = -\frac{\ln 2}{1620}$

The decay constant is $k = -\frac{\ln 2}{1620}$.

Q6b. Use your answer to determine how long it would take for the isotope to decay to 50 g. Give both an exact answer and a decimal approximation to the nearest hundredth.

Background

Topic: Exponential Decay Applications

This question tests your ability to use the decay constant to solve for the time required for a substance to decay to a given amount.

Key Terms and Formulas

• Use $y = 90e^{kt}$, with $k$ from the previous part.

• Set $y = 50$ and solve for $t$.

Step-by-Step Guidance

1. Set up the equation: $50 = 90e^{kt}$.

2. Divide both sides by 90 to isolate the exponential term.

3. Take the natural logarithm of both sides.

4. Substitute the value of $k$ from part (a) and solve for $t$.

Try solving on your own before revealing the answer!

Final Answer: $t = \frac{\ln(5/9)}{k} \approx 1042.47$ years

Exact: $t = \frac{\ln(5/9)}{k}$, with $k = -\frac{\ln 2}{1620}$. Decimal: $t \approx 1042.47$ years.

Q7. The level of sound, $\beta$, in decibels, of a sound with an intensity of $I$ is given by $\beta(I) = 10 \log_{10} \left(\frac{I}{I_0}\right)$, where $I_0 = 10^{-16}$ watts/cm$^2$. Determine the level of sound in decibels if the intensity is $I = 10^{-9}$ watts/cm$^2$.

Background

Topic: Logarithmic Scale (Decibels)

This question tests your ability to use the decibel formula to calculate sound level from intensity.

Key Terms and Formulas

• Decibel formula: $\beta = 10 \log_{10} \left(\frac{I}{I_0}\right)$

Step-by-Step Guidance

1. Substitute $I = 10^{-9}$ and $I_0 = 10^{-16}$ into the formula.

2. Compute the ratio $\frac{I}{I_0}$.

3. Take the logarithm base 10 of the ratio.

4. Multiply the result by 10 to get the decibel level.

Try solving on your own before revealing the answer!

Final Answer: 70 dB

$\beta = 10 \log_{10}(10^7) = 10 \times 7 = 70$ dB.

Q8a. Graph the following system of inequalities and label your solution set "S":

$y + 2x \geq 2$ $y - x < 4$

Background

Topic: Systems of Linear Inequalities

This question tests your ability to graph linear inequalities and find the feasible region (solution set) on the coordinate plane.

Key Terms and Formulas

• Linear inequality: An inequality involving a linear function.

• Feasible region: The set of points that satisfy all inequalities.

Step-by-Step Guidance

1. Rewrite each inequality in slope-intercept form ($y = mx + b$) if needed.

2. Graph the boundary lines for each inequality (solid for $\geq$, dashed for $<$).

3. Shade the region that satisfies each inequality.

4. The solution set "S" is where the shaded regions overlap.

Try solving on your own before revealing the answer!

Final Answer:

The solution set "S" is the region above the line $y = -2x + 2$ (including the line) and below the line $y = x + 4$ (not including the line).

Q8b. Find the maximum and minimum of the objective function $P = 2x + 4y$ given the feasible set pictured to the right.

Background

Topic: Linear Programming

This question tests your ability to find the maximum and minimum values of a linear objective function over a feasible region defined by inequalities.

Key Terms and Formulas

• Objective function: The function to maximize or minimize.

• Vertices: The maximum and minimum occur at the vertices of the feasible region.

Step-by-Step Guidance

1. Identify the vertices (corner points) of the feasible region from the graph.

2. Plug each vertex into the objective function $P = 2x + 4y$.

3. Compare the values to determine the maximum and minimum.

Try solving on your own before revealing the answer!

Final Answer:

The maximum and minimum values of $P$ occur at the vertices of the feasible region. Substitute each vertex into $P = 2x + 4y$ to find the values.

Q9. 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 $110. The company has 150 work-hours available for assembling furniture and 190 work-hours available for covering. The company wants to maximize its profit. Let$x$= the number of sofas and$y$ = the number of recliners. Write the objective function and the system of constraints. DO NOT SOLVE.

Background

Topic: Linear Programming (Formulation)

This question tests your ability to formulate a linear programming problem, including the objective function and constraints, based on a word problem.

Key Terms and Formulas

• Objective function: The function to maximize (profit).

• Constraints: Inequalities representing resource limitations.

Step-by-Step Guidance

1. Write the profit function: $P = 95x + 110y$.

2. Write the assembly time constraint: $4x + 3y \leq 150$.

3. Write the covering time constraint: $2x + 5y \leq 190$.

4. Include non-negativity constraints: $x \geq 0$, $y \geq 0$.

Try writing the constraints and objective function before revealing the answer!

Final Answer:

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

Q10a. Write the augmented matrix that is associated with the following system of equations:

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

Background

Topic: Matrices and Systems of Equations

This question tests your ability to write the augmented matrix for a system of linear equations.

Key Terms and Formulas

• Augmented matrix: A matrix that includes the coefficients and constants from a system of equations.

Step-by-Step Guidance

1. Write the coefficients of $x$, $y$, and $z$ from each equation in order.

2. Write the constants as the last column, separated by a vertical bar.

Try writing the matrix before revealing the answer!

Final Answer:

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

Q10b. Solve the above system of equations either algebraically or using matrices. Show your work neatly and clearly.

Background

Topic: Solving Systems with Matrices or Algebraic Methods

This question tests your ability to solve a system of three equations in three variables, either by elimination/substitution or by using matrix methods (such as Gaussian elimination).

Key Terms and Formulas

• Gaussian elimination: Row operations to reduce the matrix to row-echelon form.

• Back substitution: Solve for variables starting from the last equation.

Step-by-Step Guidance

1. Use row operations to eliminate variables and reduce the matrix to upper triangular (row-echelon) form.

2. Solve for one variable using the last row.

3. Substitute back into previous rows to solve for the remaining variables.

4. Write the solution as an ordered triple $(x, y, z)$.

Try solving on your own before revealing the answer!

Final Answer: $(x, y, z) = (1, 2, -1)$

Solving the system, we find $x = 1$, $y = 2$, $z = -1$.