Question

Two hot air balloons are traveling along the same path away from a town, beginning from different locations at the same time. Henry's balloon begins 20 miles from the town and is 48 miles from the town after 2 hours. The distance of Tasha's balloon from the town is represented by the function y= 10 x + 12. Which balloon was farther from the town at the beginning, and which traveled more quickly? • A. Henry's balloon was farther from the town at the beginning, and it traveled more quickly.

• B. Henry's balloon was farther from the town at the beginning, but
Tasha's balloon traveled more quickly.
c. Tasha's balloon was farther from the town at the beginning, but Henry's balloon traveled more quickly.
• D. Tasha's balloon was farther from the town at the beginning, and it traveled more quickly.

Answer 1

well, let's take a look at Henry's equation, let's make a table

`\begin{array}{ccll} \stackrel{x}{hours}&\stackrel{y}{miles}\\ \cline{1-2} 0&20\\ 2&48 \end{array}`

to get the equation of any straight line, we simply need two points off of it, so let's use those two then for Henry

`(\stackrel{x_1}{0}~,~\stackrel{y_1}{20})\qquad (\stackrel{x_2}{2}~,~\stackrel{y_2}{48}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{48}-\stackrel{y1}{20}}}{\underset{run} {\underset{x_2}{2}-\underset{x_1}{0}}} \implies \cfrac{ 28 }{ 2 } \implies 14`

`\begin{array}c \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{20}=\stackrel{m}{ 14}(x-\stackrel{x_1}{0})\implies {\Large \begin{array}{llll} y=14x+20 \end{array}}`

now, let's take a peek at Tasha's equation, hmmm y = 10x + 12

at 0 hour, namely when x = 0, Tasha's balloon was at y = 10(0) + 12 => y = 12

was 12 miles from town, whilst we know Henry's was 20 miles from town, so Henry's balloon was farther.

now, the slope in an equation, is simply the rate of change, so the larger the slope, the larger the rate of change or namely the faster it moves.

well, hell Tasha's slope is "10", whilst Henry's is "14", so Henry's balloon is going faster.