Real numbers & Pythagorean theorem
Student B is correct. √36 = 6, which is an integer, and all integers are rational numbers.
Explain why the Pythagorean Theorem only works for right triangles.
Because it is derived from the relationship between the squares of the sides in a right triangle (based on the right angle). It does not hold for non-right triangles.
Two equations are given:
y = 2x + 3
y = 2x − 5
Without graphing, determine how many solutions the system has and explain why.
No solution.
Both lines have the same slope (2) but different y-intercepts, so they are parallel and never intersect.
Simplify and explain your reasoning:
2^3 ⋅2^5
2^3+5=2^8=256
(When multiplying powers with the same base, add exponents.)
Two cubes have side lengths 3 cm and 6 cm.
Without calculating fully, determine which cube has the greater volume and explain how you know.
The cube with side length 6 cm has greater volume.
Reason: Volume depends on the cube of the side length, so doubling the side length increases volume by 2^3=8 times.
Without using a calculator, determine which is greater: √45 or 6. Explain your reasoning.
√45 is less than 6 because √36 = 6 and √49 = 7, and 45 is closer to 36 than 49, so √45 is slightly more than 6? Wait correction → √45 ≈ 6.7, so it is greater than 6. Proper reasoning: since 45 > 36, √45 > √36 = 6.
A triangle has sides 7, 9, and 12. Without finding angles, determine if it is a right triangle. Show reasoning.
Check:
7² + 9² = 49 + 81 = 130
12² = 144
Since 130 ≠ 144, it is NOT a right triangle.
A student solves the system and gets (4, 9).
x + y = 13
2x − y = −1
Explain how the student can check if the solution is correct.
Substitute (4, 9) into both equations:
4 + 9 = 13 ✔
2(4) − 9 = 8 − 9 = −1 ✔
Since it satisfies both, the solution is correct.
A student says:
(3^2)^4=3^6
Is the student correct? Explain why or why not.
Incorrect.
(3^2)^4=3^2⋅4=3^8
(When raising a power to a power, multiply exponents.)
A rectangular prism has dimensions 4, 5, and 6.
Explain how changing only one dimension (for example, doubling the height) affects the volume.
Volume = length × width × height.
Doubling one dimension doubles the volume (it scales linearly with each dimension).
A number is between √2 and √3. Is the number rational, irrational, or could it be both? Explain.
It could be both. There are both rational and irrational numbers between √2 and √3.
Find all possible integer values of x such that a triangle with sides x, 8, and 17 is a right triangle.
x² + 8² = 17²
x² + 64 = 289
x² = 225
x = 15
Solve the system using substitution and explain each step:
y = 3x − 2
2x + y = 10
2x + (3x − 2) = 10
5x − 2 = 10
5x = 12
x = 12/5
y = 3(12/5) − 2 = 36/5 − 10/5 = 26/5
Solution: (12/5, 26/5)
Simplify completely and express with positive exponents:
5^6/5^2
5^6−2=5^4=625
Two similar cylinders have a scale factor of 3 (large to small).
How many times larger is the volume of the larger cylinder? Explain your reasoning.
Volume scale factor = 3^3 = 27.
The larger cylinder has 27 times the volume.
Explain why the decimal expansion of √7 never terminates or repeats.
Because √7 is irrational, and irrational numbers have non-terminating, non-repeating decimal expansions.
A square has side length 10. Find the exact length of its diagonal and explain how you know.
Diagonal = √(10² + 10²) = √200 = 10√2
Using the Pythagorean Theorem on the right triangle formed by the diagonal.
A system is shown below:
3x + 2y = 12
6x + 4y = 30
Without solving completely, determine how many solutions the system has. Justify your reasoning clearly.
No solution.
Reasoning:
Multiply the first equation by 2 →
6x + 4y = 24
Compare with second equation:
6x + 4y = 30
Same left side but different constants → contradiction → parallel lines → no solution.
Write the number 0.00072 in scientific notation. Then explain how you know your answer is correct.
7.2×10^−4
Explanation:
Move the decimal 4 places to the right to get 7.2, so exponent is −4.
A cylinder has radius 3 cm and height 10 cm.
Another cylinder has radius 6 cm and height 10 cm.
How many times larger is the volume of the second cylinder?
Volume depends on r^2h
6^2/3^2=36/9=4
The second cylinder has 4 times the volume.
Prove or disprove: The sum of a rational number and an irrational number is always irrational.
True. If you add a rational number to an irrational number, the result remains irrational (otherwise it would contradict the definition of irrational numbers).
A triangle has side lengths √18, √32, and √50. Is it a right triangle? Justify fully.
√18 = 3√2, √32 = 4√2, √50 = 5√2
Check:
(3√2)² + (4√2)² = 18 + 32 = 50
(5√2)² = 50
A school sells tickets for a play. Student tickets cost $4 and adult tickets cost $8. A total of 50 tickets were sold, and the total revenue was $280.
Let:
x = student tickets
y = adult tickets
Equations:
x + y = 50
4x + 8y = 280
Solve:
Divide second equation by 4 →
x + 2y = 70
Subtract first equation:
(x + 2y) − (x + y) = 70 − 50
y = 20
Then:
x + 20 = 50 → x = 30
Solution: (30, 20)
Meaning:
30 student tickets and 20 adult tickets were sold.
The distance from Earth to the Sun is about 1.5×10^8 km.
The distance from Earth to the Moon is about 3.8×10^5 km.
How many times farther is the Sun than the Moon from Earth? Express your answer in scientific notation and explain your reasoning.
1.5×10^8/3.8 x 10^5 ≈0.395×103=3.95×102
So, about:
3.95×1023.95×102
Meaning: The Sun is about 395 times farther than the Moon.
A cube has volume 64 cm³.
A similar cube has volume 512 cm³.
Find the scale factor (small to large).
512/64=8
Scale factor = cube root of 8=2
Scale factor = 2