Question

the measure of angle r is $\frac{2pi}{3}$ radians. the measure of angle t is $\frac{5pi}{12}$ radians greater than the measure of angle r. what is the measure of angle t, in degrees?
a) 23
b) 120
c) 195
d) 390

$y = x^{2}-14x + 22$
the given equation relates the variables x and y. for what value of x does the value of y reach its minimum?

a small business owner budgets $2,200 to purchase candles. the owner must purchase a minimum of 200 candles to maintain the discounted pricing. if the owner pays $4.90 per candle to purchase small candles and $11.60 per candle to purchase large candles, what is the maximum number of large candles the owner can purchase to stay within the budget and maintain the discounted pricing?

$yleq x + 7$
$ygeq - 2x-1$
which point (x,y) is a solution to the given system of inequalities in the xy - plane?
a) (-14,6)
b) (0,-14)
c) (8,14)
d) (14,0)

Explanation:

First Problem (Angle - related)

Step1: Find measure of angle T in radians

First, find the measure of angle T in radians. Given angle R is $\frac{2\pi}{3}$ radians and angle T is $\frac{5\pi}{12}$ radians greater than angle R. So, $T=\frac{2\pi}{3}+\frac{5\pi}{12}$.
$T=\frac{8\pi + 5\pi}{12}=\frac{13\pi}{12}$ radians.

Step2: Convert radians to degrees

Use the conversion formula $1$ radian $=\frac{180^{\circ}}{\pi}$.
$T=\frac{13\pi}{12}\times\frac{180^{\circ}}{\pi}= 195^{\circ}$

Step1: Set up the inequality

Let $x$ be the number of small - candles and $y$ be the number of large - candles. We know $x + y\geq200$ (to maintain discounted pricing) and $4.9x+11.6y\leq2200$. Also, $x\geq0,y\geq0$. Since we want to find the maximum of $y$, we can assume $x = 200 - y$ (the minimum number of small candles to get the discounted price). Substitute $x = 200 - y$ into the cost inequality: $4.9(200 - y)+11.6y\leq2200$.

Step2: Expand and solve the inequality

Expand: $980-4.9y + 11.6y\leq2200$.
Combine like terms: $6.7y\leq2200 - 980$.
$6.7y\leq1220$.
$y\leq\frac{1220}{6.7}\approx182.09$. Since $y$ is the number of candles, the maximum integer value of $y$ is 182.

Third Problem (Quadratic Function)

Step1: Recall vertex formula for a quadratic function

For a quadratic function $y = ax^{2}+bx + c$, the x - coordinate of the vertex (where the function reaches its minimum or maximum) is given by $x=-\frac{b}{2a}$. In the function $y=x^{2}-14x + 22$, $a = 1$, $b=-14$, and $c = 22$.

Step2: Calculate the x - value of the minimum

Substitute $a = 1$ and $b=-14$ into the formula $x=-\frac{b}{2a}$.
$x=-\frac{-14}{2\times1}=7$.

Fourth Problem (Inequality Solution)

Answer:

C. 195

Second Problem (Business - Budgeting)