Middle School Math Quiz: Using Formulas

5 questions · exam conditions

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Using FormulasQuestion 1 of 5

The volume of a cylinder is given by $V = \pi r^2 h$ . If a cylindrical tank has a volume of $288\pi$ cubic inches and a height that is 2 inches more than its radius, what is the radius of the tank?

8 inches

9 inches

6 inches

12 inches

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Middle School Math Quiz

Middle School Math Quiz: Using Formulas

Practice Using Formulas in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Using Formulas, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The volume of a cylinder is given by $V = \pi r^2 h$ . If a cylindrical tank has a volume of $288\pi$ cubic inches and a height that is 2 inches more than its radius, what is the radius of the tank?

8 inches
9 inches
6 inches (correct answer)
12 inches

Explanation: Given V = πr²h = 288π and h = r + 2. Substituting: πr²(r + 2) = 288π. Dividing by π: r²(r + 2) = 288, so r³ + 2r² = 288. Rearranging: r³ + 2r² - 288 = 0. Testing r = 6: 6³ + 2(6²) = 216 + 72 = 288 ✓. We can verify: if r = 6, then h = 8, and V = π(36)(8) = 288π, which matches the given volume.

Question 2

The formula for the circumference of a circle is $C = 2\pi r$ , where $r$ is the radius. A circular track has a circumference that is 24 meters more than 4 times the radius. What is the radius of the track in meters?

$\frac{12}{\pi - 2}$
$\frac{24}{2\pi - 4}$ (correct answer)
$\frac{12}{2\pi - 4}$
$\frac{24}{\pi - 2}$

Explanation: Given that the circumference is 24 more than 4 times the radius: C = 4r + 24. Since C = 2πr, we have 2πr = 4r + 24. Solving for r: 2πr - 4r = 24, so r(2π - 4) = 24, giving r = 24/(2π - 4). Choice A is incorrect because it has (π - 2) in the denominator instead of (2π - 4), and 12 in the numerator instead of 24. Choice C has the correct denominator but wrong numerator (12 instead of 24). Choice D has the wrong denominator (π - 2) instead of (2π - 4).

Question 3

The surface area of a sphere is $SA = 4\pi r^2$ , where $r$ is the radius. If the surface area of a spherical ball is $144\pi$ square inches, what is the diameter of the ball?

12 inches (correct answer)
6 inches
18 inches
24 inches

Explanation: Given SA = 4πr² = 144π. Dividing both sides by 4π: r² = 36, so r = 6 inches. Since diameter = 2r, the diameter is 2(6) = 12 inches. Choice B gives the radius, not the diameter. Choice C (18) would correspond to r = 9, giving SA = 4π(81) = 324π ≠ 144π. Choice D (24) would correspond to r = 12, giving SA = 4π(144) = 576π ≠ 144π.

Question 4

The area of a trapezoid is given by $A = \frac{1}{2}h(b_1 + b_2)$ , where $h$ is the height and $b_1$ and $b_2$ are the parallel bases. A trapezoid has an area of 84 square centimeters, a height of 8 centimeters, and one base that is 3 centimeters longer than the other base. What is the length of the shorter base?

7.5 centimeters
8.5 centimeters
6 centimeters
9 centimeters (correct answer)

Explanation: When you encounter a word problem involving formulas, the key is to identify what you know, what you need to find, and how the given information relates to the variables in the formula. You're given the trapezoid area formula

A = \frac{1}{2}h(b_1 + b_2)

along with

A = 84

h = 8

, and the fact that one base is 3 cm longer than the other. Let's call the shorter base

x

and the longer base

x + 3

. Substituting into the formula:

84 = \frac{1}{2}(8)(x + x + 3)

Simplifying:

84 = 4(2x + 3)

Dividing by 4:

21 = 2x + 3

Subtracting 3:

18 = 2x

Therefore:

x = 9

The shorter base is 9 centimeters, making the answer D. Let's examine why the other choices are incorrect. Choice A (7.5 cm) would give you a total of the two bases as 18 cm, resulting in an area of 72 square centimeters—too small. Choice B (8.5 cm) would make the sum of bases 20 cm, yielding an area of 80 square centimeters—still too small. Choice C (6 cm) would create a sum of 15 cm for the bases, producing only 60 square centimeters—far too small. When solving formula problems with relationships between variables, always define your variables clearly based on the given relationships. Here, recognizing that "3 cm longer" means you can express both bases in terms of a single variable is the key to setting up the equation correctly.

Question 5

The area of a triangle is $A = \frac{1}{2}bh$ , where $b$ is the base and $h$ is the height. In triangle ABC, the height to the base is represented by $(x + 3)$ inches and the base is represented by $(2x - 1)$ inches. If the area is 35 square inches, what is the value of $x$ ?

4
5 (correct answer)
6
3

Explanation: Using A = (1/2)bh with A = 35, b = 2x - 1, and h = x + 3: 35 = (1/2)(2x - 1)(x + 3). Multiplying both sides by 2: 70 = (2x - 1)(x + 3). Expanding: 70 = 2x² + 6x - x - 3 = 2x² + 5x - 3. Rearranging: 2x² + 5x - 73 = 0. Using the quadratic formula: x = (-5 ± √(25 + 584))/4 = (-5 ± √609)/4. Since √609 ≈ 24.68, x ≈ (-5 + 24.68)/4 ≈ 4.92 or x ≈ (-5 - 24.68)/4 ≈ -7.42. Since x must be positive to ensure positive dimensions, x ≈ 5.