Precalculus Study Guide: Sample Test 2 (Math 1100, Spring 2026)

Q1. Express 12.5664 radians in degrees.

Background

Topic: Radian-Degree Conversion

This question tests your understanding of how to convert an angle from radians to degrees, a fundamental skill in trigonometry and precalculus.

Key formula:

$\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

Step-by-Step Guidance

1. Identify the given value: $12.5664$ radians.

2. Recall the conversion formula: $\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$.

3. Substitute $12.5664$ for radians in the formula: $\text{Degrees} = 12.5664 \times \frac{180}{\pi}$.

4. Set up the multiplication, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Final Answer: 720°

$12.5664 \times \frac{180}{\pi} = 720$

This is because $12.5664$ is approximately $4\pi$, and $4\pi$ radians equals $720$ degrees.

Q2. Express 440° in radian measure in terms of $\pi$. Simplify answer in exact form.

Background

Topic: Degree-Radian Conversion

This question tests your ability to convert degrees to radians and express the answer in terms of $\pi$.

Key formula:

$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

Step-by-Step Guidance

1. Identify the given value: $440^\circ$.

2. Recall the conversion formula: $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$.

3. Substitute $440$for degrees: $440 \times \frac{\pi}{180}$.

4. Simplify the fraction $\frac{440}{180}$ to lowest terms, but do not multiply out yet.

Try solving on your own before revealing the answer!

Final Answer: $\frac{11\pi}{9}$ radians

$440 \times \frac{\pi}{180} = \frac{11\pi}{9}$

The answer is simplified to lowest terms and left in terms of $\pi$ as required.

Q3. Convert 800 rpm (revolutions per minute) to radians per second.

Background

Topic: Angular Velocity and Unit Conversion

This question tests your ability to convert rotational speed from revolutions per minute to radians per second, using the relationship between revolutions and radians.

Key formula:

• $1 \text{ revolution} = 2\pi \text{ radians}$

• $1 \text{ minute} = 60 \text{ seconds}$

Step-by-Step Guidance

1. Start with $800$revolutions per minute ($800 \ \text{rpm}$).

2. Convert revolutions to radians: $800 \ \text{rev/min} \times 2\pi \ \text{rad/rev}$.

3. Convert minutes to seconds: $\frac{1}{60}$ minutes per second.

4. Combine the conversions: $800 \times 2\pi \div 60$ radians per second.

Try solving on your own before revealing the answer!

Final Answer: $\frac{80\pi}{3}$ radians per second

$800 \times 2\pi \div 60 = \frac{1600\pi}{60} = \frac{80\pi}{3}$ radians per second.

This is the exact value in terms of $\pi$.

Q4. If the area of a sector $A = 16.5$ m$^2$ and the radius $r = 4.02$ m, find the arc length $s$.

Background

Topic: Area and Arc Length of a Sector

This question tests your understanding of the relationship between the area of a sector, its radius, and the arc length.

Key formulas:

• $A = \frac{1}{2} r^2 \theta$ (area of a sector, $\theta$ in radians)

• $s = r\theta$ (arc length, $\theta$ in radians)

Step-by-Step Guidance

1. Write the formula for the area of a sector: $A = \frac{1}{2} r^2 \theta$.

2. Substitute the given values: $16.5 = \frac{1}{2} (4.02)^2 \theta$.

3. Solve for $\theta$ (the central angle in radians) by isolating $\theta$ in the equation.

4. Once you have $\theta$, use the arc length formula: $s = r\theta$ with $r = 4.02$.

Try solving on your own before revealing the answer!

Final Answer: $s \approx 4.11$ m

First, solve for $\theta$, then plug into $s = r\theta$ to find the arc length.

All units are in meters as required.

Q5. The paddles of a riverboat have a radius of 2.59 m and revolve at 20 rev/min. What is the speed of a tip of one of the paddles (in m/s)?

Background

Topic: Linear Speed and Circular Motion

This question tests your ability to relate angular velocity to linear speed at the edge of a rotating object.

Key formulas:

• Linear speed: $v = r\omega$

• Angular velocity: $\omega = 2\pi f$, where $f$ is frequency in revolutions per second

Step-by-Step Guidance

1. Convert revolutions per minute to revolutions per second: $20 \div 60$.

2. Calculate angular velocity: $\omega = 2\pi \times f$.

3. Use the linear speed formula: $v = r\omega$ with $r = 2.59$ m.

4. Set up the multiplication, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Final Answer: $v \approx 5.42$ m/s

After converting units and applying the formulas, the speed of the paddle tip is approximately $5.42$ m/s.

Q6. Solve $x^2 + 30 = 11x$.

Background

Topic: Quadratic Equations

This question tests your ability to solve a quadratic equation by rearranging and factoring or using the quadratic formula.

Key formula:

Standard form: $ax^2 + bx + c = 0$

Step-by-Step Guidance

1. Rewrite the equation in standard form: $x^2 - 11x + 30 = 0$.

2. Identify $a = 1$, $b = -11$, $c = 30$.

3. Factor the quadratic, or set up the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

4. Set up the factors or the formula, but do not solve for $x$ yet.

Try solving on your own before revealing the answer!

Final Answer: $x = 5$ or $x = 6$

Factoring gives $(x - 5)(x - 6) = 0$, so $x = 5$ or $x = 6$.

Q7. Factor $x^3 - 27$.

Background

Topic: Factoring Special Polynomials (Difference of Cubes)

This question tests your ability to recognize and factor the difference of cubes.

Key formula:

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Step-by-Step Guidance

1. Recognize that $x^3 - 27$ is a difference of cubes: $x^3 - 3^3$.

2. Apply the formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ with $a = x$, $b = 3$.

3. Write the factors: $(x - 3)(x^2 + 3x + 9)$.

Try solving on your own before revealing the answer!

Final Answer: $(x - 3)(x^2 + 3x + 9)$

This is the standard factorization for the difference of cubes.

Q8. Find the discriminant, $D$, of $4x^2 - 6 = 5x$.

Background

Topic: Discriminant of a Quadratic Equation

This question tests your ability to find the discriminant, which tells you about the nature of the roots of a quadratic equation.

Key formula:

Discriminant: $D = b^2 - 4ac$

Step-by-Step Guidance

1. Rewrite the equation in standard form: $4x^2 - 5x - 6 = 0$.

2. Identify $a = 4$, $b = -5$, $c = -6$.

3. Plug these values into the discriminant formula: $D = (-5)^2 - 4 \times 4 \times (-6)$.

4. Set up the calculation, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Final Answer: $121$

The discriminant is positive, so there are two real roots.