A parabolic satellite television antenna has a diameter of 16 feet and is 2 foot deep. How far is the focus from the vertex?
is the answer 8ft?
yes
Yes, the answer is 8 feet. To understand why, let's break down the steps to find the distance from the vertex to the focus of a parabolic antenna.
Step 1: Understand the properties of a parabola
A parabolic shape is defined by a particular mathematical equation, y = ax^2, where y is the height at a particular point on the parabola, x is the distance from the vertex, and a is a constant. In this case, we are given the diameter and depth of the antenna, which helps in calculating the value of a.
Step 2: Identify relevant properties of the parabolic equation
In the equation y = ax^2, the value of a can be calculated using a specific property of a parabola. The property is that the distance from the vertex to the focus of a parabola is equal to 1/4a. Therefore, we need to calculate the value of a to find the distance from the vertex to the focus.
Step 3: Calculate the value of a
Given the diameter of 16 feet, we know that the width of the parabola is 16 feet. The equation for the width of a parabola is width = 4a, where a is the constant in the equation y = ax^2. In this scenario, the width is 16 feet, so we can solve for a: 16 = 4a. Dividing both sides by 4, we get a = 4.
Step 4: Find the distance from the vertex to the focus
Now that we know the value of a is 4, we can calculate the distance from the vertex to the focus. According to the property mentioned earlier, this distance is equal to 1/4a. Plugging in the value of a, we have 1/4 * 4 = 1 foot.
So, the distance from the vertex to the focus of the parabolic satellite television antenna is 1 foot, not 8 feet.