Question

1. a triangular park has vertices a(-2,9), b(-8,-2) and c(4,3) and paths along the perimeter of the triangle. a new path is being built from vertex b to the opposite side. the path costs $12.70/meter and is being built in a way that minimizes cost (i.e., the shortest distance).

a) sketch a diagram on the grid provided. 1c
b) to minimize the cost, the path must meet the opposite side at a perpendicular angle. determine the equation of this line, in slope - intercept form. 3k
c) determine the length of the new path in meters (rounded to two decimal places). note: the equation of line ac is y=-x + 7 4a
d) determine the cost of the path (rounded to two decimal places). 1a

Explanation:

Step1: Find the slope of line AC

The equation of line AC is $y=-x + 7$, so its slope $m_{AC}=-1$.

Step2: Find the slope of the perpendicular line from B to AC

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of the line from $B(-8,-2)$ to AC be $m$. Then $m\times(-1)=-1$, so $m = 1$.

Step3: Find the equation of the line from B to AC in slope - intercept form

Using the point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(-8,-2)$ and $m = 1$, we have $y+2=1\times(x + 8)$. Simplifying gives $y=x+6$.

Step4: Find the intersection point of the two lines

We solve the system of equations

$$\begin{cases}y=-x + 7\\y=x+6\end{cases}$$

. Substitute $y$ from the second equation into the first: $x + 6=-x+7$. Solving for $x$ gives $2x=1$, so $x=\frac{1}{2}$. Then $y=\frac{1}{2}+6=\frac{13}{2}$. The intersection point $D(\frac{1}{2},\frac{13}{2})$.

Step5: Calculate the length of the new path

Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ with $(x_1,y_1)=(-8,-2)$ and $(x_2,y_2)=(\frac{1}{2},\frac{13}{2})$.
\[

$$\begin{align*} d&=\sqrt{(\frac{1}{2}+8)^2+(\frac{13}{2}+2)^2}\\ &=\sqrt{(\frac{17}{2})^2+(\frac{17}{2})^2}\\ &=\sqrt{\frac{289}{4}+\frac{289}{4}}\\ &=\sqrt{\frac{289\times2}{4}}\\ &=\frac{17\sqrt{2}}{2}\approx12.02 \end{align*}$$

\]

Step6: Calculate the cost of the path

The cost per meter is $\$13.70$. So the cost $C=13.70\times12.02 = 163.674\approx163.67$.

Answer:

a) (Sketching is a manual process and cannot be shown here in text - but plot the points $A(-2,9)$, $B(-8,-2)$ and $C(4,3)$ and draw the perpendicular from $B$ to $AC$ on the given grid)
b) $y=x + 6$
c) $12.02$ meters
d) $\$163.67$