A satellite dish in storage has parabolic cross sections and is resting on its vertex. A point on the rim is 4 ft high and is 6 ft horizontally from the vertex. How high is a point which is 3 ft horiozntally from the vertex? Draw a sketch and solve.
if we let the vertex be at (0,0),
y = 4(x/6)^2 = x^2/9
so, what's y(3)?
To solve this problem, we can use the properties of the parabolic cross-section of the satellite dish. Let's start by drawing a sketch to visualize the situation.
First, draw a horizontal line to represent the ground. Mark a point on this line to represent the vertex of the satellite dish. From this point, draw two upward-sloping lines to form the cross-section of the dish. One line should represent the point on the rim that is 6 ft horizontally from the vertex, and the other line should represent the point we want to find, which is 3 ft horizontally from the vertex. Label the height of the first point as 4 ft.
Next, we need to understand the key property of a parabolic cross-section: the distance from the vertex to the cross-section is proportional to the square of the horizontal distance from the vertex. In other words, we have a relationship like this:
d^2 = k * h,
where d is the horizontal distance from the vertex, h is the height above the vertex, and k is a constant of proportionality.
In our case, we know that when d = 6 ft, h = 4 ft. Let's use this information to find the value of k.
(6 ft)^2 = k * 4 ft,
36 ft^2 = 4k,
k = 36 ft^2 / 4 ft,
k = 9 ft.
Now that we have the value of k, we can use it to find the height of the point that is 3 ft horizontally from the vertex.
(3 ft)^2 = 9 ft * h,
9 ft = 9 ft * h,
h = 1 ft.
Therefore, the height of the point that is 3 ft horizontally from the vertex is 1 ft.
By applying the key property of parabolic cross-sections and using the given information, we were able to determine the height of the point of interest.