A landscaper is designing a garden with hedges through which a straight path will lead from the exterior of the garden to the interior. If the polar coordinates of the endpoints of the path are (20, 90°) and (10, 150°), where r is measured in feet, what is the equation for the path?
To find the equation for the path, we can use the polar to rectangular conversion formulas:
x = r * cos(θ)
y = r * sin(θ)
Using the given polar coordinates:
For the first endpoint:
r1 = 20
θ1 = 90°
For the second endpoint:
r2 = 10
θ2 = 150°
Converting the polar coordinates to rectangular coordinates:
For the first endpoint:
x1 = 20 * cos(90°) = 0
y1 = 20 * sin(90°) = 20
For the second endpoint:
x2 = 10 * cos(150°) = -5√3
y2 = 10 * sin(150°) = -5
Now, let's find the equation of the line passing through these two points. The slope of the line can be found using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values:
m = (-5 - 20) / (-5√3 - 0)
= (-25) / (-5√3)
= 5 / √3
We can think of the equation in the form of y = mx + b, where b is the y-intercept.
Substituting the coordinates of one endpoint (0, 20):
20 = (5 / √3) * 0 + b
b = 20
Therefore, the equation for the path is:
y = (5 / √3) * x + 20
To find the equation for the path, we need to express the points in Cartesian coordinates (x, y).
Let's convert the polar coordinates to Cartesian coordinates:
The first point, (r1, θ1) = (20, 90°), can be converted as follows:
x1 = r1 * cos(θ1) = 20 * cos(90°) = 20 * 0 = 0
y1 = r1 * sin(θ1) = 20 * sin(90°) = 20 * 1 = 20
Similarly, the second point (r2, θ2) = (10, 150°) can be converted as follows:
x2 = r2 * cos(θ2) = 10 * cos(150°)
y2 = r2 * sin(θ2) = 10 * sin(150°)
Now that we have the Cartesian coordinates of the points, we can find the equation of the straight path connecting them.
The equation of a straight line is usually given by the formula y = mx + c, where m is the slope and c is the y-intercept.
To find the slope of the line, we can use the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values, we get:
m = (y2 - y1) / (x2 - x1)
m = (10 * sin(150°) - 20) / (10 * cos(150°) - 0)
After simplifying, we get the value of m.
Now that we have the slope, we can find the y-intercept (c) using one of the points.
Substituting the values of one point (x, y) into the equation:
y = mx + c
We can solve for c by substituting the values of x, y, and m into the equation and solving for c.
Finally, we have the slope (m) and the y-intercept (c), so we can write the equation of the path as:
y = mx + c