Solving this problem actually requires us the use of the distance formula of point to a line.
The formula is:
distance = | a x + b y + c | / sqrt (a^2 + b^2)
So we are given the equation:
y = 2 x + 4
rewriting:
y – 2 x – 4 = 0 --> a = -2, b = 1, c = -4
We are also given the points:
(-4, 11) = (x, y)
Using the distance formula at points (x, y):
distance = | -2 * -4 + 1 * 11 + -4 | / sqrt [(- 2)^2 + (1)^2]
distance = 15 / sqrt (5)
distance = 6.7
So the tree is about 6.7 ft away from the zip line.
The question is an illustration of distance between a point and a line.
The distance between the zip line and the tree is 6.70 feet
The zip line equation is given as:
[tex]\mathbf{y = 2x +4}[/tex]
The coordinates of the tree is given as:
[tex]\mathbf{(x,y) = (-4,11)}[/tex]
The distance (d) between an equation and a point is:
[tex]\mathbf{d = \frac{|ax + by + c|}{\sqrt{a^2 + b^2}}}[/tex]
Where:
[tex]\mathbf{ax +by + c = 0}[/tex]
Rewrite [tex]\mathbf{y = 2x +4}[/tex] as [tex]\mathbf{ax +by + c = 0}[/tex]
[tex]\mathbf{2x - y + 4 = 0}[/tex]
By comparison:
[tex]\mathbf{a =2, b = -1, c = 4}[/tex] and [tex]\mathbf{(x,y) = (-4,11)}[/tex]
So, we have:
[tex]\mathbf{d = \frac{|2 \times -4 -1 \times 11 + 4|}{\sqrt{(2)^2 + (-1)^2}}}[/tex]
[tex]\mathbf{d = \frac{|-15|}{\sqrt{5}}}[/tex]
Evaluate square root
[tex]\mathbf{d = \frac{|-15|}{2.24}}[/tex]
Remove absolute bracket
[tex]\mathbf{d = \frac{15}{2.24}}[/tex]
[tex]\mathbf{d = 6.70}[/tex]
Hence, the distance between the zip line and the tree is approximately 6.70 feet
Read more about distance at:
https://brainly.com/question/11558698
