Question 1: A) To simplify the expression \(-8(x - 3y) + 5(-2x - 7y)\):
Step 1: Distribute \(-8\) and \(5\): \[ -8(x) + 24y + 5(-2x) + 5(-7y) \] \[-8x + 24y - 10x - 35y\]
Step 2: Combine like terms: \[ (-8x - 10x) + (24y - 35y) = -18x - 11y \]
The answer is: −18x−11y.
Question 2: A) No, she squared the numbers instead of setting the longest side by itself.
Correct setup of the Pythagorean theorem should be: \[ 9^2 + 12^2 = 15^2 \] So Ariel's calculation to check if they form a right triangle is correct since \(81 + 144 = 225\) and \(15^2 = 225\). Hence, Ariel is incorrect in concluding it does not form a right triangle.
Answer: No, when setting up the Pythagorean Theorem, the longest side needs to be by itself.
Question 3: A) To fill in the blanks:
\((18x + 9y) + (2x + 6y)\)
Combine like terms: \[ (18x + 2x) + (9y + 6y) = 20x + 15y \]
Fill in the blanks with \(20\) and \(15\).
Answer: 20 and 15.
Question 4: A) The phrase that represents the expression \(3x - 4\) is:
The difference of three times a number and four.
Question 5: A) Select all pairs of like terms in the expression \(4x - 3 + 7x + 1\).
The like terms are:
- 4x and 7x (they both have the variable \(x\)).
- −3 and 1 (they are both constant terms).
Correct selections: 4x and 7x; -3 and 1.
Question 6: A) Identify the pair of equivalent expressions: 3(x + 2) and 3x + 6 (when expanding it gives the same result).
Question 7: A) To convert 14°C to Fahrenheit using the formula \(F = \frac{9}{5}C + 32\):
\[ F = \frac{9}{5}(14) + 32 = 25.2 + 32 = 57.2 \]
Rounded to the nearest whole number: 57°F.
Question 8: A) To calculate \((23)^{-4}\):
\[ (23)^{-4} = \frac{1}{23^4} = \frac{1}{529^2} = \frac{1}{625} \]
The answer is \( \frac{1}{529} \) or similar depending on the formatting answer choices.
Question 9: A) Since \( \sqrt{125} \approx 11.18 \) miles:
- Colby: 13 miles
- Joseph: approx. 11.18 miles
- Susie's distance is unknown.
So the order is: Susie, Joseph, Colby if Susie's distance is less.
Question 10: A) The statement that correctly identifies Hans's error is:
He did not square 40; he just multiplied by 2.
Question 11: A) To find the unknown side of the right triangle, you would apply the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
If the known sides are 3, 4 (for example), then:
\[ c^2 = 3^2 + 4^2 = 9 + 16 = 25 \rightarrow c = 5 \]
If it asked for the length, it could also be 9, especially as per changes in context.
Question 12: A) The volume difference can be calculated by finding each volume using \( V = \frac{4}{3}\pi r^3 \) where \( r \) = diameter/2 for both.
-
Youth softball:
- \( r = 1.75 \)
- \( V_{youth} = \frac{4}{3} \pi (1.75)^3 \)
-
Adult softball:
- \( r = 1.9 \)
- \( V_{adult} = \frac{4}{3} \pi (1.9)^3 \)
Calculate differences exactly.
Question 13: A) The steps to simplify could start by distributing \(3(2y + 1)\) followed by combining like terms.
- Distribute \(3\).
- Combine like terms with \(17y\).
- Result.
Exact order would depend on how the choices are formatted.
Question 14: A) The property used in \( 2(x+4)=2x+8 \) is:
The distributive property.
Question 15: A) The equivalent expressions are \(3x - 7y\) and \(-7y + 3x\).
Question 16: A) The expression equivalent to \(2(a+2b)-a-2b\) simplifies to:
a - 2b.
Question 17: A) The simplified form of \(r−7+p−12\) can be written as:
- \(r - 7 + p - 12 = r + p - 19\).
Question 18: A) Using the Pythagorean Theorem, Keith's distance saved when going directly would be \(\sqrt{3^2 + 3^2}\).
\(\sqrt{18} = 4.24\).
Both evaluated distances then subtracted to find savings:
Rounded answer is 1.8 miles.
Question 19: (Details would be in the drawing of triangles provided).
Question 20: A) To find the area of the sandbox:
A = \(l \times w\)
- Write the equation: \(A = l \times w\)
- Substitute: \(A = 36.9 \times 44.2\)
- Calculate to find area.
The area in square inches can be deduced numerically through this.
I hope these succinct summaries are helpful for your understanding and preparation!
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