Question

If an arch is in the shape of a parabola and it spans 80 ft with a maximum height of 20 feet what is the equation for the parabola and what is it's height from the center?

Answer 1

The equation for the parabolic arch is y = 20x - (1/80)x² and the height from the center is 780 ft.

Step-by-step explanation:

The equation for the parabolic arch can be found by using the formula y = ax + bx², where a and b are constants. In this case, we know that the arch spans 80 ft and has a maximum height of 20 ft. To find the equation, we need to find the values of a and b.

Since the maximum height occurs at the vertex of the parabola, we can plug in the coordinates of the vertex (0, 20) to get the equation 20 = 0a + 0b². Since a and b are both multiplied by 0, this equation tells us that a = 20.

Next, we can use the fact that the arch spans 80 ft to find the value of b. Plugging in the coordinates of one of the endpoints (40, 0), we get the equation 0 = 40a + 40b². Substituting the value of a we found earlier, we can solve for b and find that b = -1/80.

Therefore, the equation for the parabolic arch is y = 20x - (1/80)x², and the height from the center can be found by plugging in the x-coordinate of the center into the equation. Since the arch spans 80 ft, the center would be at x = 40, so plugging in x = 40, we get y = 20(40) - (1/80)(40)² = 800 - 20 = 780 ft.

Answer 2

Step-by-step explanation:

Given that:

• Spans = 80ft

• Maximum height = 20ft

As we know that the Parabola is of type :

`(x - x_(0)) ^(2)` = -4a(y -
`y_(0)` ) where:
`x_(0) y_(0)` is the origin of the line so we have:

`x_(0) = 0` and
`y_(0) = 20`

<=>
`x^(2)` = -4a(y-20)

at y = 0, we have:

`x^(2)` = -4a(0-20)

<=>
`x^(2) = 80a`

<=> x = ±
`√(80a)`

But we know that 2x= 80 <=> x= 40, so:

40 =
`√(80a)`

<=> 1600 = 80a

<=> a = 20

The equation for the parabola is:
`x^(2) = 80(y-20)`

It's height from the center at x= 0 => y= 20