Question

A parabolic lattice arch is 8 feet high at the vertex. At a height of 4 feet, the width of the lattice is 4 ft. How wide is the lattice arch at ground level?

Answer 1

The width of the lattice arch at ground level is 2 feet.

Step-by-step explanation:

To find the width of the lattice arch at ground level, we can set up a proportion using the given information. Let's let x represent the width at ground level.

Since the height at 4 feet is half of the total height, the width at that height should also be half.

So, we have the proportion: 4/8 = x/w, where w is the width at ground level.

Cross multiplying, we get 4w = 8x.

Dividing both sides by 4, we get w = 2x. So, the width at ground level is twice the width at a height of 4 feet.

Given that the width at a height of 4 feet is 4 feet, we can substitute this value in the equation to find the width at ground level. 4 = 2x.

Dividing both sides by 2, we get x = 2. So, the width of the lattice arch at ground level is 2 feet.

Answer 2

The arch is in the shape of an inverted parabola. The equation of the arch is found as follows:

`y=-ax^(2)+8`
Substituting the values at a height of 4 ft, we get:
4 = -4a + 8
a = 1
At ground level y = 0:

`0=-x^(2)+8`

`x=+- √(8)`
therefore the width of the arch at ground level is:

`2 √(8) \ feet`