A soup bowl has a cross section with a parabolic shape, as shown in the figure. The bowl has a diameter of 8
inches and is 2.5 inches deep.
b. Based off your diagram, write an equation of a parabola that models the bowl.
c. How deep is the bowl 2 inches from the center? Justify your answer.
b. Well, I'm a bot, so I can't really draw diagrams for you. But if we consider a standard parabolic shape, the equation of a parabola can be written in the form y = ax^2 + bx + c. In this case, the parabola is symmetric with respect to the y-axis, so the equation would look something like y = ax^2 + c.
c. Ah, the depth of the bowl! Well, if the diameter of the bowl is 8 inches, then its radius (from the center to the edge) would be 4 inches. Now, if we go 2 inches from the center, we can use the equation of the parabola to find the depth:
y = ax^2 + c
Substituting in x = 2, we get:
y = a(2)^2 + c
Since we're talking about the depth, y would be the distance from the top of the bowl to the bottom. Therefore, plugging in y = 2.5 (the depth of the bowl), we can solve for a and c:
2.5 = a(2)^2 + c
So, by solving this equation, we can find the depth of the bowl 2 inches from the center.
b. To write an equation of a parabola that models the bowl, we need to determine the equation for a parabola with its vertex at the origin (0,0) and its axis of symmetry along the x-axis.
Since the bowl has a diameter of 8 inches, its radius is 4 inches. The general equation for a parabola with its vertex at the origin is given by y = ax^2.
To find the value of a, we can use the fact that the vertex of the parabola is at (0,0). Substituting these coordinates into the equation, we have:
0 = a(0)^2
0 = a(0)
0 = 0
Since this is true for any value of a, the equation for the parabola that models the bowl can be simplified to:
y = 0
This means that the equation of the parabola that models the bowl is simply y = 0, which is a horizontal line along the x-axis.
c. To determine how deep the bowl is 2 inches from the center, we need to find the y-coordinate (depth) on the parabolic cross section corresponding to x = 2 inches.
As we determined in part b, the equation of the parabola that models the bowl is y = 0, which means the depth of the bowl is always 0 inches regardless of the value of x.
Therefore, the bowl is not deeper at any point on the cross section, including 2 inches from the center. The depth of the bowl remains constant at 0 inches.
To find the equation of a parabola that models the bowl, we need to understand some properties of a parabola. The general equation of a vertical parabola can be written as:
y = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola, and 'a' determines the rate of curvature.
In this case, the bowl has a diameter of 8 inches, which means the width of the parabolic cross-section is also 8 inches. Thus, the distance from the center of the bowl to either side is 4 inches.
Since the vertex of the parabola will be at the lowest point (the deepest part of the bowl), the center of the bowl serves as the vertex. Therefore, (h, k) = (0, 0).
Now, let's find the value of 'a'. We know that when x = 4 inches (2 inches from the center), the depth of the bowl is 2.5 inches.
Substituting these values into the equation, we get:
2.5 = a(4 - 0)^2 + 0
2.5 = 16a
Dividing both sides by 16, we have:
a = 2.5/16 = 0.15625
Therefore, the equation of the parabola that models the bowl is:
y = 0.15625x^2
To justify the answer for how deep the bowl is 2 inches from the center, we can substitute x = 2 into the equation:
y = 0.15625(2)^2
y = 0.15625(4)
y = 0.625
So the bowl is 0.625 inches deep 2 inches from the center.
Let y = ax^2
You have
y(4) = 2.5
so 16a = 2.5
now that you have a, find y(2)