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A satellite dish has a cross-section in the shape of the parabola x² = 16y, with lengths in…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A satellite dish has a cross-section in the shape of the parabola $x^2 = 16y$ , with lengths in centimetres and the vertex at the bottom. The receiver is placed at the focus. Find three things. First, the height of the receiver above the bottom of the dish. Second, the width of the dish at that height. Third, the depth of the dish at a place $12$ centimetres from the axis.

A satellite dish drawn as the upward-opening parabola x squared equals 16y in centimetres, from x = -16 to 16. The receiver is marked as a dot at (0, 4), the focus, 4 centimetres above the bottom of the dish. A red horizontal line through the receiver spans the dish from (-8, 4) to (8, 4).

Need a reminder? The key ideaShow

The parabola $y^2 = 4ax$ has its focus at $(a, 0)$ and its directrix $x = -a$ . The latus rectum has length $4a$ , and the focal distance of $(x_1, y_1)$ is $x_1 + a$ . The point $(at^2, 2at)$ is on it for every $t$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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