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Chapter 8: Problem 26

Students noticed that the path of water from a water fountain seemed to form a parabolic arc. They set a flat surface at the level of the water spout and measured the maximum height of the water from the flat surface as 8 inches and the distance from the spout to where the water hit the flat surface as 10 inches. Construct a function model for the stream of water.

Short Answer

Expert verified
The function model is \(y = -\frac{8}{25}(x - 5)^2 + 8\).

Step by step solution

01

Understand the Problem

Identify the important values given: the maximum height of the water is 8 inches, and the horizontal distance from the spout to where the water hits the flat surface is 10 inches.
02

Set the Vertex

Since the maximum height is 8 inches and occurs at the midpoint of the 10-inch distance, place the vertex of the parabola at \(x = 5\), 8 inches above the flat surface. Thus, the vertex is (5,8).
03

General Form of Parabola

Use the standard form of a parabola that opens downwards: \(y = a(x-h)^2 + k\), where (h,k) is the vertex. In this case, \(h = 5\) and \(k = 8\), so the equation becomes \(y = a(x - 5)^2 + 8\).
04

Find the Value of 'a'

Use the point where the water hits the flat surface. At \(x = 0\), \(y = 0\). Substitute \(x = 0\) and \(y = 0\) into the equation \(0 = a(0 - 5)^2 + 8\) to solve for 'a': \ 0 = a(25) + 8 -8 = 25a a = -\frac{8}{25}.
05

Construct the Function

Substitute the value of 'a' back into the equation. The function model for the stream of water is \(y = -\frac{8}{25}(x - 5)^2 + 8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertex form of a parabola
A parabolic function can be expressed in various forms, one of which is the vertex form. This form is especially useful for easily identifying the vertex of the parabola. The vertex form of a parabola is given by the equation:
\( y = a(x - h)^2 + k \)
Here:
  • 'a' determines the width and the direction of the parabola's opening.
  • (h, k) is the vertex of the parabola.
This form makes it simple to see where the parabola reaches its peak or its lowest point. In our exercise, the vertex is at (5, 8), where the maximum height of the water fountain's stream is 8 inches, occurring at 5 inches from the spout. Substituting h and k into the vertex form, we get:
\( y = a(x - 5)^2 + 8 \)
We'll now need the value of 'a' to complete the function model.
maximum height
The maximum height of a parabolic trajectory, like our water fountain's arc, happens at the vertex of the parabola. Understand that in a real-world problem such as this, we need precise measurements to identify this point.
In the given exercise, the maximum height is measured as 8 inches. This height occurs exactly at the midpoint of the water stream's path from the spout to where it lands, understood as 10 inches total.
This means the maximum height of the water is 8 inches when it is halfway through its path, or 5 inches from the spout.
To find the mathematical model, we placed the vertex at (5,8). This point is pivotal because it provides the highest point on the parabola. Therefore, our equation becomes centered around these values using the vertex form.
distance and midpoint
Understanding distance and midpoint is key to solving problems involving parabolas in real life. The distance from the spout to where the water hits the surface is 10 inches. This is the total horizontal distance.
The midpoint, which is crucial for finding the vertex in our parabolic function, is found by dividing the distance by 2. Therefore:
\[ \text{Midpoint} = \frac{10}{2} = 5 \text{ inches} \]
So, at 5 inches, the water reaches its maximum height. This is reflected in the vertex form as (5,8).
Placing the parabola's vertex at (5, 8), we use this information along with the point where the water hits the ground to form our equation. By substituting into the vertex form of a parabola:
We get \( y = a(x - 5)^2 + 8 \), where 'a' is calculated by using another point on the parabola, in this case, (0,0).

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