Question

1. marco designs a spinner wheel that has exactly 4 sections: red, blue, green, and yellow. he wants the spinner wheel to have a 25% chance of landing on each section. he spins the wheel 500 times. the results of the spins are shown in this table.

spinner wheel section	number of times the spinner lands in each section
blue	165
green	130
yellow	125

based on the results in this table, one of the following changes would be the best fix. which one?
a. he should decrease the area of the red section by increasing the area of the blue section.
b. he should increase the area of the red section by decreasing the area of the blue section.
c. he should increase the area of the red section by decreasing the area of any of the other three sections.
d. he should decrease the area of the blue section, and then it does not matter which section’s area is increased.

1. in $\triangle abc$, $\angle a$ and $\angle c$ are congruent, and the measure of $\angle b$ is $143.6^\circ$. what is the measure of $\angle a$?

f. $18.2^\circ$
g. $36.4^\circ$
h. $71.8^\circ$
j. $143.6^\circ$

1. which of the following expressions is equivalent to $x^2 - x - 30$?

a. $(x + 3)(x - 10)$
b. $(x + 6)(x - 5)$
c. $(x - 6)(x + 5)$
d. $(x - 15)(x - 15)$

1. which of the following matrices is equal to $5\

$$\begin{bmatrix}-4 & 2 \\\\ 0 & -5\\end{bmatrix}$$

$?
f. $\

$$\begin{bmatrix}-20 & -15\\end{bmatrix}$$

$
g. $\

$$\begin{bmatrix}-10 \\\\ -25\\end{bmatrix}$$

$
h. $\

$$\begin{bmatrix}1 & 7 \\\\ 5 & 0\\end{bmatrix}$$

$
j. $\

$$\begin{bmatrix}-20 & 10 \\\\ 0 & -25\\end{bmatrix}$$

$

Explanation:

Question 3

First, calculate the expected number of times each section should be landed on: \( 500 \times 25\% = 125 \) times.

• Red: 80 (less than 125)
• Blue: 165 (more than 125)
• Green: 130 (close to 125)
• Yellow: 125 (exact).

To make the chance 25% for each, the red section (which landed too few times) needs a larger area (so it's more likely to be landed on), and the blue section (which landed too many times) needs a smaller area. So we should increase the area of red by decreasing blue's area. Option B says this. Option A does the opposite. Option C says decrease any other section, but green and yellow are close or exact, so decreasing them isn't right. Option D says it doesn't matter which is increased, but it does (should increase red). So B is correct.

Step1: Recall triangle angle sum

The sum of angles in a triangle is \( 180^\circ \). Let \( \angle A = \angle C = x \) (since they are congruent).

Step2: Set up equation

\( x + x + 143.6^\circ = 180^\circ \)

Step3: Solve for x

Combine like terms: \( 2x + 143.6^\circ = 180^\circ \)
Subtract \( 143.6^\circ \): \( 2x = 180^\circ - 143.6^\circ = 36.4^\circ \)
Divide by 2: \( x = \frac{36.4^\circ}{2} = 18.2^\circ \)

Step1: Factor the quadratic

We need to factor \( x^2 - x - 30 \). Find two numbers that multiply to -30 and add to -1.

Step2: Identify the numbers

The numbers are -6 and +5? Wait, no: \( -6 \times 5 = -30 \), but \( -6 + 5 = -1 \)? Wait, no: \( 6 \) and \( -5 \): \( 6 \times (-5) = -30 \), and \( 6 + (-5) = 1 \). Wait, no, the middle term is -x, so the numbers should be -6 and +5? Wait, no: let's check the options.
Option B: \( (x + 6)(x - 5) = x^2 -5x +6x -30 = x^2 +x -30 \) (no). Wait, wait, maybe I messed up. Wait, \( x^2 -x -30 \). Let's find two numbers: product -30, sum -1. So -6 and +5: \( -6 \times 5 = -30 \), \( -6 + 5 = -1 \). Wait, but \( (x - 6)(x + 5) = x^2 +5x -6x -30 = x^2 -x -30 \). Wait, option C is \( (x - 6)(x + 5) \). Wait, let's check the options again:

• A: \( (x + 3)(x - 10) = x^2 -10x +3x -30 = x^2 -7x -30 \)
• B: \( (x + 6)(x - 5) = x^2 -5x +6x -30 = x^2 +x -30 \)
• C: \( (x - 6)(x + 5) = x^2 +5x -6x -30 = x^2 -x -30 \)
• D: \( (x - 15)(x - 15) = x^2 -30x +225 \)

Ah, so option C is correct. Wait, earlier mistake: the numbers are -6 and +5, so factored as \( (x - 6)(x + 5) \), which is option C.

Step1: Find factors of -30

We need two numbers \( a \) and \( b \) such that \( a \times b = -30 \) and \( a + b = -1 \).

Step2: Determine \( a \) and \( b \)

The numbers are \( -6 \) and \( 5 \) (since \( -6 \times 5 = -30 \) and \( -6 + 5 = -1 \)).

Step3: Write the factored form

Thus, \( x^2 -x -30 = (x - 6)(x + 5) \), which is option C.

Answer:

B. He should increase the area of the red section by decreasing the area of the blue section.

Question 4