Problem 56 - Precalculus with Calculus Previews

Question

Suppose that two towers of a suspension bridge are \(350 \mathrm{ft}\) apart and the vertex of the parabolic cable is tangent to the road midway between the towers. If the cable is \(1 \mathrm{ft}\) above the road at a point \(20 \mathrm{ft}\) from the vertex, find the height of the towers above the road.

Step-by-Step Solution

Answer

The towers are 76.5625 ft tall above the road.

Step 1: Identify the Parabola Equation

The cable can be modeled as a parabola because it is described as a parabolic cable. We place the vertex of the parabola at the origin \(0, 0\) for simplicity. Therefore, the equation of the parabola is of the form \(y = ax^2\).

Step 2: Determine the Constant 'a'

We know the cable is \(1 \, \text{ft}\) above the road at \(x = 20 \, \text{ft}\) away from the vertex. Substitute these values into the equation \(1 = a(20)^2\). Solve for \(a\):\\[a = \frac{1}{400}\].

Step 3: Calculate the Required Height

We are asked to find the height of the towers, which are located at \(x = -175 \, \text{ft}\) and \(x = 175 \, \text{ft}\) from the vertex due to symmetry (half of \(350 \, \text{ft}\)). Substitute these x-values into the equation \(y = \frac{1}{400}x^2\) to find the height of the cable at the towers.\For \(x = 175\):\\[y = \frac{1}{400}(175)^2 = \frac{30625}{400} = 76.5625\].

Step 4: Interpret the Final Result

The calculated y-value \(76.5625 \, \text{ft}\) represents the height of the cable above the road at the points where the towers are located. Therefore, the towers are \(76.5625 \, \text{ft}\) tall above the road.

Key Concepts

Vertex of a Parabola

When dealing with parabolas, the vertex is one of the most crucial points to identify. The vertex represents the minimum or maximum point of the parabola, depending on its orientation. In the problem of the suspension bridge, the vertex is crucial because it is where the cable, which follows a parabolic path, is closest to the road. It's halfway between the towers and situated perfectly at the lowest point. By placing the vertex at the origin,

• we simplify calculations,
• make it easy to use symmetry, and
• develop the equation of the parabola.

In mathematical terms, the origin corresponds to the vertex that gives us a starting point for defining the parabola's equation of the form \( y = ax^2 \). Understanding the vertex helps us determine how the parabola is aligned in relation to other objects, in this case, the road and towers.

Height Calculation

Calculating the height of specific points on the parabola, such as the towers, involves substituting certain values into the parabola's equation. The towers are located symmetrically relative to the vertex, making it easier to compute. In this case, the towers are 175 feet from the vertex in either direction, being half the total distance of 350 feet between the towers. Using the known equation \( y = \frac{1}{400}x^2 \),

• we insert \( x = 175 \) to find the height of the cable above the road at the towers.
• This helps us calculate the precise height by the relation \( y = \frac{30625}{400} \).
• The result, \( y = 76.5625 \) feet, gives the height directly above the road.

This height implies how the cable stretches over the space between the towers and gives insight into how much support the towers need.

Equation of a Parabola

To describe any parabolic motion, such as the cable between the towers, we use the equation of a parabola. A standard form of this equation is \( y = ax^2 + bx + c \). For our bridge problem, by placing the vertex at the origin, we simplify the equation to \( y = ax^2 \). Because the problem indicates the cable is a parabolic shape, identifying the correct equation allows us to model the cable accurately.

• We initially solve for the constant \( a \) using known conditions, like how far the cable is from the road at certain points.
• For instance, using the condition that the cable is 1 ft above the road at 20 ft from the vertex, we determine \( a \) as \( \frac{1}{400} \).
• This value of \( a \) helps define the curve's steepness or spread.

Understanding the equation's role highlights which points on the parabola we can measure, helping calculate necessary features like height or distance above the road efficiently.