To determine the width of the arch at its base, we need to identify the points where the arch intersects the [tex]\( x \)[/tex]-axis, also known as the x-intercepts or the zeroes of the equation.
The given equation for the height of the arch is:
[tex]\[ y = -x^2 - 6x + 16 \][/tex]
To find the x-intercepts, we set [tex]\( y \)[/tex] equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ -x^2 - 6x + 16 = 0 \][/tex]
Solving this quadratic equation for [tex]\( x \)[/tex] will give us the zeroes. By factoring or using the quadratic formula, we find the roots:
[tex]\[ x = -8 \][/tex]
and
[tex]\[ x = 2 \][/tex]
These are the points where the arch meets the x-axis. To find the width of the arch at its base, we calculate the distance between these two x-intercepts.
The distance (or width of the arch) is given by the absolute difference between these x-values:
[tex]\[ \text{Width of the arch} = |2 - (-8)| = |2 + 8| = 10 \][/tex]
Thus, the width of the arch at its base is 10 units. This corresponds to "the difference between the zeroes."
Therefore, the correct option is:
- the difference between the zeroes
