To solve this problem, we need to determine the span of a semiellipse-shaped arch given specific dimensions.
### Given:
1. The height of the arch at the center (major axis length, 2a) is 35 feet. Hence, the semi-major axis [tex]\( a \)[/tex] is:
[tex]\[ a = \frac{35}{2} = 35 \text{ feet} \][/tex]
2. The height of the arch at a point 26 feet away from the center (minor axis length, 2b) is 16 feet above ground.
3. The distance from the center where the height is measured is 26 feet.
### Step-by-Step Solution:
#### Step 1: Understanding the ellipse formula
For an ellipse, the equation is:
[tex]\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \][/tex]
where [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are the coordinates of any point on the ellipse, [tex]\( a \)[/tex] is the semi-major axis, and [tex]\( b \)[/tex] is the semi-minor axis.
#### Step 2: Rearrange the equation to solve for [tex]\( b^2 \)[/tex]
We have:
[tex]\[ x = 26 \text{ feet} \][/tex]
[tex]\[ y = 16 \text{ feet} \][/tex]
[tex]\[ a = 35 \text{ feet} \][/tex]
Plug these into the ellipse equation and solve for [tex]\( b^2 \)[/tex]:
[tex]\[ \frac{26^2}{b^2} + \frac{16^2}{35^2} = 1 \][/tex]
[tex]\[ \frac{676}{b^2} + \frac{256}{1225} = 1 \][/tex]
[tex]\[ \frac{676}{b^2} + 0.2093877551 = 1 \][/tex]
[tex]\[ \frac{676}{b^2} = 1 - 0.2093877551 \][/tex]
[tex]\[ \frac{676}{b^2} = 0.7906122449 \][/tex]
[tex]\[ b^2 = \frac{676}{0.7906122449} \][/tex]
[tex]\[ b^2 \approx 854.5923633 \][/tex]
#### Step 3: Calculate [tex]\( b \)[/tex] (semi-minor axis)
[tex]\[ b = \sqrt{854.5923633} \][/tex]
[tex]\[ b \approx 29.2334118 \text{ feet} \][/tex]
#### Step 4: Determine the span of the bridge
The span of the bridge is the full length of the minor axis, or [tex]\( 2b \)[/tex]:
[tex]\[ \text{Span} = 2b \][/tex]
[tex]\[ \text{Span} = 2 \times 29.2334118 \][/tex]
[tex]\[ \text{Span} \approx 58.4668235 \text{ feet} \][/tex]
Finally, rounding to two decimal places:
[tex]\[ \text{Span} \approx 58.47 \text{ feet} \][/tex]
Therefore, the span of the bridge should be approximately 58.47 feet.
Thus, the correct answer is:
[tex]\[ \boxed{58.47 \text{ ft}} \][/tex]
