Question

1. the function (d(x)=\frac{x}{4}+2.5) represents the number of miles a boat travels (x) minutes after 8 a.m. complete the sentence. for each box, choose the letter before the number or phrase. (d(30)=\begin{cases}a&110\\b&10\\c&7.5end{cases}) and represents (\begin{cases}d&\text{the number of minutes traveled after 8:00 a.m.}\\e&\text{the number of minutes traveled at 8:00 a.m.}\\f&\text{the number of miles traveled after 8:00 a.m.}\\g&\text{the number of miles traveled at 8:00 a.m.}end{cases}) 7. the arc - length ((l_a)) of a sector of a circle with angle measure (x) (in degrees) and radius (r), is given by the equation (l_a=\frac{x}{360}cdot2pi r). solve the equation for the radius of the sector, (r). (\begin{cases}a&r = \frac{360cdot l_a}{2pi x}\\b&r=\frac{360cdot l_a}{2pi}\\c&r=\frac{2pi x}{360}cdot l_a\\d&r=\frac{l_a-\frac{x}{360}}{2pi}end{cases})

Explanation:

Step1: Evaluate $D(30)$ for the first - part

Given $D(x)=\frac{x}{4}+2.5$, substitute $x = 30$ into the function.
$D(30)=\frac{30}{4}+2.5$
$=\frac{30}{4}+\frac{10}{4}=\frac{30 + 10}{4}=\frac{40}{4}=10$

Step2: Interpret the meaning of $D(30)$

Since $D(x)$ represents the number of miles a boat travels $x$ minutes after 8:00 a.m., $D(30)$ represents the number of miles traveled 30 minutes after 8:00 a.m.

Step3: Solve the arc - length formula for $r$

Given $L_a=\frac{x}{360}\cdot2\pi r$.
First, multiply both sides of the equation by $\frac{360}{x}$ to isolate $r$:
$\frac{360}{x}L_a = 2\pi r$.
Then, divide both sides by $2\pi$:
$r=\frac{360\cdot L_a}{2\pi x}$

Answer:

1. B. 10; G. the number of miles traveled 30 minutes after 8:00 a.m.
2. A. $r=\frac{360\cdot L_a}{2\pi x}$