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Answer:
Part A) The length of a straight line between the school and the Fire Station is
Part B) The length of a straight line between the school and the hospital is
Step-by-step explanation:
<u><em>The complete question in the attached figure</em></u>
Let
The positive x-coordinates ----> East
The positive y-coordinates ----> North
The negative x-coordinates ----> West
The negative y-coordinates ----> South
take the point A (0,0) as the Middle School (reference point)
Part A) we have
Town Hall is located 4.3 miles directly east of the Middle School
so
The coordinates of Town Hall are B(4.3,0)
The Fire Station is located 1.7 miles directly North of Town Hall
so
The coordinates of Fire Station are C(4.3,1.7)
What is the length of a straight line between the school and the Fire Station?
Remember that
the formula to calculate the distance between two points is equal to
we have
A(0,0) and C(4.3,1.7)
substitute
Part B) we have that
The hospital is 3.1 miles west of the fire station
so
The coordinates of the hospital are D(4.3-3.1,1.7)
D(1.2,1.7)
What is the length of a straight line between the school and the hospital?
Remember that
the formula to calculate the distance between two points is equal to
we have
A(0,0) and D(1.2,1.7)
substitute
To write the system we need the slope of each line and at least one point on the line. The two lines to consider will be the lines connecting the location of each plane to the airport they are flying to. It is also worth noting that the coordinates of the airport represent the point of intersection of the two lines and thus the solution to the system.
1. slope of the line connecting airplane one and the airport: m = 2 you can see this clearly if you graph the two points. From airplane 1 location we rise 8 units and move to the right 4 units to get to the airport. Slope is defined as rise over run: so 8 divided by 4 = 2(the slope) Now substitute the slope and the point (2,4) into point-slope form of a line:
y - 4 = 2(x -4) the standard form of this equation is 2x - y = 0
2. slope of the line connecting airplane 2 and the airport: m = -
To find this slope, simply observe the vertical change of down 3 and a horizontal shift of right 9 from the airport to airplane 2. Now substitute this slope and and the point (15,9) into point-slope form of a line:
y - 9 =
(x - 15) the standard form of this equation is:
x + 3y = 42
Let's write the system:
2x - y = 0
x + 3y = 42
Multiply the first equation by 3 to get the new system
6x - 3y = 0
x + 3y = 42 add these two equations to get an equation in terms of x
7x = 42 thus x = 6 and substituting this value into 2x - y = 0 we see y = 12
In other words, we have proven that the location of the airport is in fact the solution to our system.
PS: You just have to do a little algebra to get from point-slope form of the two equations to standard form. I did not show this process, but if you need it just let me know... thanks
1. slope of the line connecting airplane one and the airport: m = 2 you can see this clearly if you graph the two points. From airplane 1 location we rise 8 units and move to the right 4 units to get to the airport. Slope is defined as rise over run: so 8 divided by 4 = 2(the slope) Now substitute the slope and the point (2,4) into point-slope form of a line:
y - 4 = 2(x -4) the standard form of this equation is 2x - y = 0
2. slope of the line connecting airplane 2 and the airport: m = -
y - 9 =
x + 3y = 42
Let's write the system:
2x - y = 0
x + 3y = 42
Multiply the first equation by 3 to get the new system
6x - 3y = 0
x + 3y = 42 add these two equations to get an equation in terms of x
7x = 42 thus x = 6 and substituting this value into 2x - y = 0 we see y = 12
In other words, we have proven that the location of the airport is in fact the solution to our system.
PS: You just have to do a little algebra to get from point-slope form of the two equations to standard form. I did not show this process, but if you need it just let me know... thanks