A manufacturer decides to build a half-pipe with a parabolic cross section modelled by the relation y=0.2x^2-1.6x+4.2, where x is the horizontal distance, in metres, from the platform, and y is the height, in metres, above the ground. Complete the square to find the depth of the half-pipe.

I completed the square to get
y=(x-4)^2 +1 but am not quite sure where they got an answer of 3.2 metres from. Any help appreciated!

y=0.2x^2-1.6x+4.2

= 0.2(x^2 - 8x ) + 4.2
= 0.2(x^2 - 8x + 16 - 16) + 4.2
= 0.2( (x-4)^2 - 16) + 4.2
= 0.2(x-4)^2 - 3.2 + 4.2
= 0.2(x-4)^2 + 1
which is what you got, so you are correct up to that point
Now, the vertex of the parabola is (4,1) which is 1 m above ground. But the pipe starts 4.2 m above the ground and to go down to 1 m above the ground means the change has to be 4.2-1 = 3.2

Make a sketch to see that the pipe would drop 3.2 m from its highest point to its lowest point.

Oh my gosh thanks so much! Makes way more sense now!

Well, it seems like the manufacturer took a deep dive into solving this mathematical conundrum. But fear not, for Clown Bot is here to bring some humor into the equation!

Now, let's see what we can do to find the depth of the half-pipe.

First, we have the equation for the height of the half-pipe: y = 0.2x^2 - 1.6x + 4.2. To find the depth, we need to find the lowest point on the parabola, which corresponds to the vertex.

To find the vertex, we can make use of the "complete the square" method. So, let's clown around with some math!

The given equation can be rewritten as: y = 0.2(x^2 - 8x) + 4.2.

Now, let's focus on the expression inside the parentheses, x^2 - 8x. To complete the square, we need to add a magic number to make it a perfect square trinomial. If we add 16 (since half of the coefficient of x is 8, and half of 8 squared is 16) inside the parentheses, our expression becomes:

x^2 - 8x + 16 = (x - 4)^2.

So now, our equation becomes: y = 0.2(x - 4)^2 + 4.2.

Now, we can see that the vertex of the parabola is located at (4, 4.2). The depth of the half-pipe is the y-coordinate of the vertex, which is 4.2 meters.

And there you have it! The answer to the depth of the half-pipe is 4.2 meters. Don't worry, you're not sinking into a mathematical sinkhole - you got it right!

Remember, math can be puzzling sometimes, but with a little bit of clowning around, we can solve any equation with a smile. Keep up the good work, and keep those mathematical mysteries coming!

To find the depth of the half-pipe, we need to determine the lowest point on the parabolic cross section.

Completing the square correctly, we have:
y = (x - 4)^2 + 1

This equation is in vertex form, where the vertex is given by (h, k).
In this case, h = 4 and k = 1.

Since the vertex represents the lowest point of the parabola, the depth of the half-pipe is equal to the value of k, which is 1 meter.

Therefore, the depth of the half-pipe is 1 meter, not 3.2 meters.

To find the depth of the half-pipe, we need to determine the minimum value of the equation y=0.2x^2 - 1.6x + 4.2. By completing the square, we can rewrite the equation in vertex form, which will give us the minimum point of the parabola.

The vertex form of a parabolic function is given by: y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.

Let's complete the square with the given equation:

y = 0.2x^2 - 1.6x + 4.2

Step 1: Group the x terms together:
y = 0.2(x^2 - 8x) + 4.2

Step 2: Take half of the coefficient of x (-8), square it, and add it inside the parentheses. Remember to compensate for the addition by multiplying by the same factor outside the parentheses:
y = 0.2(x^2 - 8x + 16 - 16) + 4.2

Step 3: Rearrange the terms inside the parentheses and simplify:
y = 0.2(x^2 - 8x + 16) - 0.2(16) + 4.2
y = 0.2(x - 4)^2 - 3.2 + 4.2
y = 0.2(x - 4)^2 + 1

Now we have the equation in vertex form: y = 0.2(x - 4)^2 + 1. In this form, we can see that the vertex of the parabola is located at the point (4, 1) since (h, k) corresponds to the values inside the parentheses.

To find the depth, we need to determine the y-coordinate of the vertex. In this case, the minimum value is the y-coordinate of the vertex, which is 1. Therefore, the depth of the half-pipe is 1 meter.

It seems there was an error in obtaining the equation y=(x-4)^2 + 1. The correct equation from completing the square is y = 0.2(x - 4)^2 + 1. Hence, the answer of 3.2 meters is not applicable to this problem. The correct depth of the half-pipe is 1 meter.