/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 56 Monthly normal rainfall data \((... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 56

Monthly normal rainfall data \((x, y)\) for Portland, Oregon, are \((4,2.47),(7,0.58),(8,1.07),\) where \(x\) represents time in months (with \(x=1\) representing January) and \(y\) represents rainfall in inches. Find the values of \(a, b,\) and \(c\) rounded to 2 decimal places such that the equation \(y=a x^{2}+b x+c\) models this data. According to your model, how much rain should Portland expect during September? (Source: National Climatic Data Center)

Short Answer

Expert verified
Portland should expect roughly 2.12 inches of rain in September.

Step by step solution

01

- Set up the system of equations

We have three data points, and our model is a quadratic equation \( y = ax^2 + bx + c \). Substitute the data points into this equation to form a system of equations. We have: For \((4,2.47)\): \[ 16a + 4b + c = 2.47 \] For \((7,0.58)\): \[ 49a + 7b + c = 0.58 \] For \((8,1.07)\): \[ 64a + 8b + c = 1.07 \] These are our three equations.
02

- Solve the system of equations

We have three equations: 1. \( 16a + 4b + c = 2.47 \)2. \( 49a + 7b + c = 0.58 \)3. \( 64a + 8b + c = 1.07 \)To solve this system, you can use substitution or elimination methods. First, subtract equation 1 from equation 2, and equation 2 from equation 3 to eliminate \(c\). Then solve for \(a\) and \(b\) with the remaining simpler system.
03

Step 2.1 - Elimination of variable c

Subtract equation 1 from equation 2: \[ (49a + 7b + c) - (16a + 4b + c) = 0.58 - 2.47 \] Which simplifies to: \[ 33a + 3b = -1.89 \] Subtract equation 2 from equation 3: \[ (64a + 8b + c) - (49a + 7b + c) = 1.07 - 0.58 \] Which simplifies to: \[ 15a + b = 0.49 \] Now, solve the two equations: 1. \( 33a + 3b = -1.89 \)2. \( 15a + b = 0.49 \)For \(a\) and \(b\).
04

- Solve for a and b

Solve the two-variable system:From equation 2: \[ b = 0.49 - 15a \] Substitute \(b\) in equation 1:\[ 33a + 3(0.49 - 15a) = -1.89 \]Which simplifies to:\[ 33a + 1.47 - 45a = -1.89 \] This further simplifies to:\[ -12a = -3.36 \] So, \( a = 0.28 \). Substitute \(a\) back into \( b = 0.49 - 15a \): \[ b = 0.49 - 15(0.28) \] \[ b = 0.49 - 4.2 \] \[ b = -3.71 \].
05

- Solve for c

Use the value of \( a = 0.28 \) and \( b = -3.71 \) in any original equation to find \(c\). Using equation 1:\[ 16(0.28) + 4(-3.71) + c = 2.47 \] \[ 4.48 - 14.84 + c = 2.47 \] \[ c = 12.83 \] Thus, the coefficients are \(a = 0.28\), \(b = -3.71\), and \(c = 12.83\).
06

- Calculate rainfall for September

Substitute \(x = 9\) (for September) into our quadratic equation, \(y = 0.28x^2 - 3.71x + 12.83\): \[ y = 0.28(9)^2 - 3.71(9) + 12.83 \] Calculate \(9^2 = 81\), so: \[ y = 0.28(81) - 3.71(9) + 12.83 \] \[ y = 22.68 - 33.39 + 12.83 \] \[ y = 2.12 \] Portland should expect approximately 2.12 inches of rain in September.

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Key Concepts

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

System of Equations
When dealing with data modeling, especially in the context of quadratic equations, we often need to figure out unknown variables by using a system of equations. Here, each equation in the system represents a condition that must be met, and all of them together define the relationships between the variables.
  • Given multiple data points, we use these to set individual equations.
  • Each point plugged into the general form of the quadratic model \( y = ax^{2} + bx + c \) helps to form these equations.
In our exercise, we used the points \((4, 2.47), (7, 0.58), (8, 1.07)\) to create three equations. In a system of equations, each equation corresponds to a point, ensuring the quadratic fits all points in the best possible way.
Solving Equations
Once you have a system of equations, the next step is to solve them. This involves finding the exact values of the unknown variables that satisfy all equations in the system. Common techniques are substitution and elimination.
  • Substitution: Solve one equation for one variable and substitute into another equation to simplify the system.
  • Elimination: Adjust the equations to eliminate one variable, making it easier to find the values for the others.
In this problem, elimination was used to simplify solving the three equations by eliminating the constant term \( c \). Two of the original equations were subtracted from each other to gradually isolate one variable at a time.
Quadratic Models
Quadratic models play a crucial role in data modeling when the data may have a parabolic trend. The general form \( y = ax^2 + bx + c \) helps capture curvature patterns in data, which are not possible using linear models.
  • Coefficient \( a \): Determines the curvature direction (upwards or downwards) of the parabola.
  • Coefficient \( b \): Controls the tilt or slope of the parabola, affecting how data increases or decreases.
  • Constant \( c \): Directly affects the parabola's vertical position along the y-axis.
In this exercise, the calculated coefficients \( a = 0.28, b = -3.71, \) and \( c = 12.83 \) form the quadratic model that most accurately fits the historical rainfall data for the given months.
Data Modeling
Data modeling is an essential aspect when interpreting data points to reveal underlying patterns or predictions. With quadratic models, we can account for more complex relationships in data than with simple linear models.
  • Utilizing Data Points: Values like \((4, 2.47), (7, 0.58), (8, 1.07)\) inform how the model is shaped and adjusted.
  • Prediction: Once parameters are determined, we can input new values to predict outcomes for months not directly measured.
For instance, in this exercise, finding the expected rainfall for September or any other month involves plugging into the model \( y = 0.28x^2 - 3.71x + 12.83 \). Through this model, despite having limited past data, we can estimate future outcomes.

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Most popular questions from this chapter

Solve each system. To do so, you may want to let \(a=\frac{1}{x}\) (if \(x\) is in the denominator) and let \(b=\frac{1}{y}\) (if \(y\) is in the denominator.) $$ \left\\{\begin{array}{c} {x+\frac{2}{y}=7} \\ {3 x+\frac{3}{y}=6} \end{array}\right. $$

Solve each system of equations. $$ \left\\{\begin{array}{r} {x-3 y=-5.3} \\ {6.3 x+6 y=3.96} \end{array}\right. $$

Solve each system of equations by the elimination method. See Examples 7 through 10. $$ \left\\{\begin{array}{r} {-2 x+3 y=0} \\ {2 x+6 y=3} \end{array}\right. $$

Add the equations. See Section 4.1. $$ \begin{aligned} x+4 y-5 z &=20 \\ 2 x-4 y-2 z &=-17 \end{aligned} $$

Solve each system of equations. $$ \left\\{\begin{array}{l} {4 x-1.5 y=10.2} \\ {2 x+7.8 y=-25.68} \end{array}\right. $$
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