Will you please check my work for me.
1)The equation of a parabola is shown.
y=1/14X^2. What are the coordinates of the focus?
(0,-3.5) <----
(0,-4)
(0,7)
2)The equation x2 + (y + 3)2 = 36 models the boundary on a local map for which Darren can hear his friend Tom on his two-way radio when Darren is at home. How far (in miles) can Tom walk from Darren's home and still be heard?
3 miles
6 miles<----
2 miles
12 miles
3)A plane intersects only one nappe of a double-napped cone. It is neither perpendicular to the cone's axis nor parallel to its generating line. Which conic section is formed?
point
circle
ellipse <-----
parabola
1. Y = (1/14)x^2.
V(h,k)
h = Xv = -b/2a = 0 / (1/7) = o.
k = Yv = (1/14)*0^2 = 0.
4a = 4(1/14) = 4/14 = 2/7.
1/4a = 7/2.
V(0,0), F(0,y).
VF = y-0 = 1/4a = 7/2
y-0 = 7/2
Y = 7/2 = 3.5.
F(0,3.5).
1) To find the coordinates of the focus of a parabola, we need to use the standard form of a parabola equation: y = (1/4p)x^2. Here, p represents the distance from the vertex to the focus. In this case, the given equation is y = (1/14)x^2.
Comparing it with the standard form, we can see that p is equal to 14. The coordinates of the focus are then given by (0, p). Substituting the value of p, we get (0, 14). Therefore, the correct answer is not among the provided options.
2) The equation given, x^2 + (y + 3)^2 = 36, represents a circle centered at (-0, -3) with a radius of 6. The distance Tom can walk from Darren's home and still be heard is equal to the radius of the circle, which is 6 miles. Therefore, the correct answer is 6 miles.
3) A plane that intersects only one nappe (branch) of a double-napped cone, without being perpendicular to the cone's axis or parallel to its generating line, forms an ellipse. Therefore, the correct answer is an ellipse.