Question

reasoning with parallel and perpendicular lines
madison is a developer, and her group is planning some additions to a new community development. currently there is just one straight road that passes through the development: moonbeam drive. the main highway is represented by the x - axis on the graph. north is at the top of the graph, and east is at the right.
write the function, s(x), that describes the path of sundance avenue on the map: s(x) =
madisons group will build a road that is perpendicular to both sundance avenue and moonbeam drive. the slope of the line representing this road on the map will be
select the function madisons group should choose to build the road perpendicular to moonbeam drive and sundance avenue:

Explanation:

Step1: Find the slope - intercept form of Moonbeam Drive

The two points on Moonbeam Drive are \((0,16)\) and \((20,0)\). The slope \(m_1\) of a line passing through \((x_1,y_1)\) and \((x_2,y_2)\) is \(m_1=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 16}{20-0}=-\frac{4}{5}\). Using the slope - intercept form \(y = mx + b\) with the point \((20,0)\), we have \(0=-\frac{4}{5}\times20 + b\), so \(b = 16\). The equation of Moonbeam Drive is \(y=-\frac{4}{5}x + 16\).

Step2: Find the equation of Sundance Avenue

We are given that the slope of Sundance Avenue is \(m =-\frac{4}{5}\), and assume it passes through the point \((0,4)\) (from the graph). Using the slope - intercept form \(y=mx + b\), substituting \(m =-\frac{4}{5}\) and \(b = 4\), the function \(s(x)=-\frac{4}{5}x+4\).

Step3: Find the slope of the perpendicular road

If two lines with slopes \(m_1\) and \(m_2\) are perpendicular, then \(m_1\times m_2=- 1\). Since the slope of Sundance Avenue and Moonbeam Drive is \(m =-\frac{4}{5}\), the slope of the line perpendicular to them is \(m_2=\frac{5}{4}\).

Step4: Find the equation of the perpendicular road

Assuming the perpendicular road passes through a point \((x_0,y_0)\) (not given, but using the slope - intercept form \(y=mx + b\) with \(m=\frac{5}{4}\), a general form of the function for the perpendicular road is \(y=\frac{5}{4}x + b\) (where \(b\) depends on the y - intercept of the actual line).

Answer:

1. \(s(x)=-\frac{4}{5}x + 4\)
2. \(\frac{5}{4}\)
3. \(y=\frac{5}{4}x + b\) (where \(b\) is a real - valued y - intercept)