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.
To determine the quadratic function that can be used to model the shape of a qulliq in vertex form, we need to find the equation of a parabola.
The vertex form of a quadratic function is given by: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
In this case, the parabolic bowl shape of the qulliq can be represented by a quadratic function using the vertex form.
Given that the length of the bowl is 30 cm and the height is 9 cm, we know that the vertex will be at the point (15, 9) (since the vertex is at the midpoint of the parabola).
Let's substitute the vertex coordinates into the vertex form equation:
y = a(x - 15)^2 + 9
Now, we need to find the value of 'a' to complete the equation.
To find the value of 'a', we can use another known point on the parabola. One known point is (0, 0), since the parabola starts from the origin.
Substitute this point into the equation and solve for 'a':
0 = a(0 - 15)^2 + 9
0 = a(225) + 9
0 = 225a + 9
225a = -9
a = -9/225
a = -1/25
Now we can plug in this value of 'a' into the equation:
y = (-1/25)(x - 15)^2 + 9
Therefore, the quadratic function that can be used to model the shape of a qulliq in vertex form is:
y = (-1/25)(x - 15)^2 + 9.
To determine the quadratic function that models the shape of a qulliq, we need to analyze its dimensions and the shape it forms. The qulliq is described as a parabolic bowl, which means it has a quadratic shape.
The general form of a quadratic function is given by:
f(x) = ax^2 + bx + c
In this case, let's assume the bottom of the qulliq is the x-axis and the sides of the qulliq define the shape. The vertex form of a quadratic function is given by:
f(x) = a(x - h)^2 + k
To determine the vertex form of the quadratic function, we need to find the values of a, h, and k. Let's use the given dimensions of the qulliq to find these values.
The length of the bowl is 30 cm, which means the x-coordinate of the vertex is at x = (30/2) = 15 cm.
The height of the bowl is 9 cm, which gives us the y-coordinate of the vertex.
The vertex form of the quadratic function is f(x) = a(x - h)^2 + k. Plugging in the values, we have:
f(x) = a(x - 15)^2 + k
To find the value of a, we need another point on the graph. Let's use the bottom edges of the qulliq, which are symmetrically at x = 0 and x = 30. The y-coordinate at these points is 0 since the bottom of the qulliq is the x-axis.
Using the point (0, 0), we have:
0 = a(0 - 15)^2 + k
0 = 225a + k
Using the point (30, 0), we have:
0 = a(30 - 15)^2 + k
0 = 225a + k
Now we have a system of equations:
225a + k = 0
225a + k = 0
Subtracting the second equation from the first, we get:
0 - 0 = (225a + k) - (225a + k)
0 = 0
This means that a can have any value since the equations are identical.
Therefore, the quadratic function that models the shape of a qulliq in vertex form is:
f(x) = a(x - 15)^2 + k
where a can be any real number and k is the y-coordinate of the vertex (which can also be any real number).