the dome over a town hall has a parabolic shape. the dome measures 48 m across and rises 12m at the centre. A vertical column needs to be attached to the dome at a point that is 4m away from its rim. How tall is the dome at this point? Also, what is the equation that models the shape of this dome?
x^2 means x squared, or x².
If you just started with parabolas, it won't hurt to say what you've learned so far.
Let's start over:
We first assume the vertex of the parabola is at x=0.
We know that y=0 at x=±24, or
y=a(x-24)(x+24).
We also know that y=12 at x=0, i.e.
12=a(0-24)(0+24)=-576
so a=12/(-576)=-1/48
The equation of the parabola is therefore:
y=(x-24)(x+24)/48
Can you then continue?
Correction to one of the above lines:
12=a(0-24)(0+24)=-576a
To find how tall the dome is at a point 4m away from its rim, we can use the equation of a parabola. A parabola can be represented by the equation:
y = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola and 'a' represents the coefficient that determines the shape of the parabola.
In this case, the dome has a width of 48m, so its diameter is also 48m. Therefore, the radius is half of the diameter, and we have a radius of 24m. The height at the center of the dome is 12m, which represents the vertex of the parabola. So, the vertex is located at (0, 12).
Next, we need to find the value of 'a' to complete the equation. To do this, we can use the known vertex and another point on the parabola, which is 4m away from the rim. We can use the coordinates of this point (4, y) and substitute them into the equation:
y = a(x - 0)^2 + 12
y = a(x^2) + 12
To find 'a', we can substitute the coordinates (24, 0) into the equation:
0 = a(24^2) + 12
0 = a(576) + 12
0 = 576a + 12
576a = -12
a = -12/576
a = -1/48
Now that we have the value of 'a', we can substitute it back into the equation to find the height at a point 4m away from the rim:
y = (-1/48)(x^2) + 12
y = (-1/48)(4^2) + 12
y = (-1/48)(16) + 12
y = -1/3 + 12
y = 35/3
Therefore, the dome is approximately 11.67m tall at a point 4m away from its rim.
The equation that models the shape of this dome is: y = (-1/48)(x^2) + 12
Take the vertex of the parabola at x=0.
Then the equation of the parabola (opening downwards) is y=12-ax^2 where a is a constant to be found.
We know that y=0 at x=±24 (half span), so
0=12-a(24^2)
=>
a=12/576=1/48
The equation of the parabola is therefore
y=12-x^2/48.
I will let you finish the problem.