A qulliq is an oil lamp that functions as the main source of light and heat for an igloo in the Inuit culture. It is made of soapstone shaped as a parabolic bowl that contains seal blubber for the oil. It is lined with Arctic wool for the wick. If the length of the bowl is 30cm and the height is 9cm determine the quadratic function that can be used to model the shape of a qulliq. Express the function in vertex form.
I assume the bowl is circular, with a parabolic cross-section. If we let the vertex be at (0,0) then we have
y = kx^2
where
y(15) = 9
so,
y = 1/25 x^2
Well, I must say this qulliq question has certainly sparked my interest! Let's shed some light on it, shall we?
To find the quadratic function that models the shape of a qulliq, we can use the vertex form of a quadratic function, which is given by:
f(x) = a(x - h)^2 + k
In this case, we can think of the parabolic bowl as a cross-section of the qulliq. Since the length of the bowl is 30cm and the height is 9cm, we can identify the vertex of the parabola as (h, k) = (15, 9). Voilà!
Now, all we need to do is determine the value of "a" to complete the quadratic function. To do this, we can use another point on the parabolic bowl. Let's take a closer look at the highest point.
Since the highest point of the parabolic bowl is the vertex itself, we know that this point lies on the graph of the quadratic function. So, we can substitute (x, y) = (15, 9) into the equation to solve for "a":
9 = a(15 - 15)^2 + 9
9 = a(0) + 9
9 = 9
Turns out the value of "a" doesn't matter in this case! The quadratic function that models the shape of a qulliq, expressed in vertex form, is:
f(x) = 9
So, no matter how you shape it, the qulliq remains a shining example of a quadratic function with a constant value.
Hopefully, that illuminates your question!
To model the shape of a qulliq, we can use a quadratic function in vertex form, which is given by:
f(x) = a(x - h)^2 + k
where (h, k) represents the vertex of the quadratic function.
In this case, we are given the length, which represents the horizontal distance, and the height, which represents the vertical distance.
First, let's find the coordinates of the vertex. In the given scenario, the length of the bowl is 30cm, so the maximum horizontal distance is half of it, which is 15cm. The height of the bowl is 9cm, which is the vertical distance from the vertex to the x-axis.
Therefore, the vertex of the parabolic bowl is (h, k) = (15, 9).
Substituting the vertex into the vertex form equation, we get:
f(x) = a(x - 15)^2 + 9
Now, let's solve for the value of 'a', which determines the shape and direction of the quadratic function.
To do this, we need another point on the parabola. Let's use the highest point of the bowl, which is one of the endpoints of the length, as (30, 0).
Substituting this point into the equation, we get:
0 = a(30 - 15)^2 + 9
0 = a(15)^2 + 9
0 = 225a + 9
-9 = 225a
a = -9/225
a = -1/25
Finally, substituting 'a' back into the equation, we get the quadratic function in vertex form for the shape of the qulliq:
f(x) = (-1/25)(x - 15)^2 + 9
So, the quadratic function that can be used to model the shape of a qulliq is f(x) = (-1/25)(x - 15)^2 + 9.
To determine the quadratic function that models the shape of a qulliq, we need to identify the parabolic shape of the bowl and express it in vertex form.
The vertex form of a quadratic function is given by the equation: f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
In this case, the bowl of the qulliq can be visualized as a parabolic bowl shape. To establish the vertex form, we need to find the coordinates of the vertex by determining the values of h and k.
Given that the length of the bowl is 30 cm and the height is 9 cm, we can consider the vertex to be at the center point, where the length is divided evenly in half. Thus, the x-coordinate of the vertex is h = 30/2 = 15 cm.
To determine the y-coordinate of the vertex, we need to find the depth of the bowl. Since the bowl is shaped as a parabolic bowl, the depth can be considered as the height. Therefore, the y-coordinate of the vertex is k = 9 cm.
Now, substituting the values of h and k into the vertex form equation, we have:
f(x) = a(x - 15)^2 + 9
The quadratic function that models the shape of a qulliq in vertex form is given by f(x) = a(x - 15)^2 + 9.