Question

REASONING A truck that is 11 feet tall and 7 feet wide is traveling under an arch. The arch can be modeled by \(y=-0.0625 x^2+1.25 x+5.75\), where \(x\) and \(y\) are measured in feet. a. Will the truck fit under the arch? Explain. b. What is the maximum width that a truck 11 feet tall can have and still make it under the arch? c. What is the maximum height that a truck 7 feet wide can have and still make it under the arch?

Short answer

a. The truck will fit under the arch if the width of the truck is less or equal to the difference of the two solutions \(x_1, x_2\) obtained from solving the quadratic equation with \(y = 11\).\nb. The maximum width a truck of 11 feet can have is given by the difference \(x_2 - x_1\).\nc. The maximum height a truck of width 7 feet can have is given by \(y\) obtained from the equation of the arch by substititing \(x = 7\).

Step-by-step solution

Find the width for a truck of 11 feet height

To find out whether the truck will fit under the arch, we need to compare the width of the truck with the width of the arch at the height of the truck. The height of the truck is 11 feet. So set \(y = 11\) in the equation for the arch: \(11=-0.0625 x^2+1.25 x+5.75\). Solve this quadratic equation to obtain the values of \(x\) which gives two points on the archway where the height is equal to the truck's height.

Compare the obtained width with the width of the truck

When you solve the quadratic equation, you get two solutions: \(x_1, x_2\). The distance between these two solutions gives the maximum width of a truck that can pass under the arch at the height of 11 feet. If the width of the truck is less than or equal to this calculated width, then the truck can pass under the arch.

Find the maximum height for a truck of 7 feet width

To find the maximum height a truck of 7 feet width can have and still make it under the arch, substitute \(x = 7\) into the equation for the arch, i.e., \(y=-0.0625 * 7^2+1.25 * 7+5.75\). Solve this equation to obtain the value of \(y\), which is the maximum height that a truck can have to pass under the arch.

Key concepts

Quadratic Modeling

Quadratic modeling is a form of mathematical representation that uses quadratic equations to describe real-world situations. In our scenario, the arch's shape is represented by the quadratic equation \(y=-0.0625 x^2+1.25 x+5.75\).

The equation translates physical attributes of the arch into a mathematical form, which allows us to predict and calculate various properties such as the maximum width and height that a truck can be to pass under the arch. By inputting values into this equation, we can simulate the behavior of trucks as they interact with the arch. This kind of modeling is essential for solving practical problems in engineering, architecture, and many other fields.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. A quadratic equation is typically in the form \(ax^2 + bx + c = 0\), and its solutions are found using a variety of methods like factoring, completing the square, or applying the quadratic formula.

In the context of our exercise, setting \(y = 11\) and solving the resulting equation helps find the points where a truck that is 11 feet tall will just fit under the arch. This involves rearranging the equation and finding the values of \(x\) that satisfy it. These solutions, \(x_1\) and \(x_2\), are crucial for determining the feasibility of the truck's passage under the arch.

Function Maximization

Function maximization is the process of finding the maximum point on a graph, which in many real-world problems corresponds to an optimum value of interest. The maximum or minimum of a quadratic function is found at its vertex, the peak or valley of its parabolic shape.

For finding the maximum width and height for the truck under the arch, the quadratic equation provides an easy way to calculate these extremal values. To maximize the truck's height for a given width, or vice versa, you identify the maximum value that \(y\) (height) can reach for a given \(x\) (width), or find the maximum spread between two points on the arch (the distance between \(x_1\) and \(x_2\)) for a given height.

Mathematical Reasoning

Mathematical reasoning involves the logical deduction of conclusions from premises known or assumed to be true. It's the backbone of solving problems efficiently and correctly. With the truck and arch problem, reasoning enables us to deduce whether the truck can fit through the arch.

It requires setting up the appropriate equations, solving them, and interpreting the results in the context of the question. For instance, once we calculate the maximum truck width at a certain height, mathematical reasoning helps us to conclude whether our truck, with its specific measurements, can pass under the arch. This solid reasoning is essential, especially when these calculations impact real-life decisions and designs.