Problem 90 - Algebra A Combined Function

Question

The relationship between body weight and the Recommended Dietary Allowance (RDA) for vitamin A in children up to age 10 is modeled by the quadratic equation \(y=0.149 x^{2}-4.475 x+406.478\) where \(y\) is the RDA for vitamin A in micrograms for a child whose weight is \(x\) pounds. (Source: Based on data from the Food and Nutrition Board, National Academy of Sciences-Institute of Medicine, 1989\()\) a. Determine the vitamin A requirements of a child who weighs 35 pounds. b. What is the weight of a child whose \(\mathrm{RDA}\) of vitamin A is 600 micrograms? Round your answer to the nearest pound.

Step-by-Step Solution

Answer

a. 431.928 micrograms; b. 54 pounds.

Step 1: Understanding the Given Information

We are given the quadratic equation \( y = 0.149x^2 - 4.475x + 406.478 \), where \( y \) represents the RDA for vitamin A in micrograms, and \( x \) represents the weight of a child in pounds. Our tasks are to find the RDA when the weight is given and to determine the weight when the RDA is known.

Step 2: Calculating Vitamin A RDA for 35 Pounds

For Part (a), we need to substitute \( x = 35 \) into the quadratic equation \( y = 0.149x^2 - 4.475x + 406.478 \). This results in \( y = 0.149 (35)^2 - 4.475 (35) + 406.478 \). Calculate \( 0.149 (35)^2 = 182.075 \), \( -4.475 (35) = -156.625 \), and finally add them to 406.478. Solve to find \( y \).

Step 3: Simplify and Solve for RDA

Performing the calculations, we have \( y = 182.075 - 156.625 + 406.478 \). Adding these, \( y = 431.928 \). Hence, the RDA for vitamin A for a child weighing 35 pounds is approximately 431.928 micrograms.

Step 4: Setting Up the Equation for Given RDA

For Part (b), given \( y = 600 \), substitute it in the equation: \( 600 = 0.149x^2 - 4.475x + 406.478 \). Rearrange it to form a standard quadratic equation \( 0.149x^2 - 4.475x + 406.478 - 600 = 0 \). This simplifies to \( 0.149x^2 - 4.475x - 193.522 = 0 \).

Step 5: Applying the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 0.149 \), \( b = -4.475 \), and \( c = -193.522 \). Calculate \( b^2 - 4ac = 4.475^2 + 4 \times 0.149 \times 193.522 \) to evaluate the discriminant.

Step 6: Solving the Quadratic Equation

Calculate \( 4.475^2 = 20.025625, 4 \times 0.149 \times 193.522 = 115.361816 \). Adding these gives the discriminant \( 135.387441 \). Take the square root \( \sqrt{135.387441} \approx 11.637 \). Now compute both roots using the quadratic formula.

Step 7: Calculating the Roots

The roots are \( x = \frac{4.475 \pm 11.637}{2 \times 0.149} \). Calculate the two possible solutions: Root 1 is \( x = \frac{4.475 + 11.637}{0.298} \approx 54.27 \) and Root 2 is \( x = \frac{4.475 - 11.637}{0.298} \approx -24.05 \). Since weight cannot be negative, we consider only the positive root.

Step 8: Rounding to the Nearest Pound

The weight of the child whose RDA of vitamin A is 600 micrograms is approximately 54 pounds after rounding 54.27 to the nearest whole number.

Key Concepts

Recommended Dietary Allowance (RDA)

The Recommended Dietary Allowance, or RDA, is a guideline provided to ensure people get the necessary nutrients to maintain health. For children, especially those up to age 10, it is crucial to get the right amount of vitamins such as Vitamin A.
Vitamin A is essential for vision, immune function, and growth. The RDA represents a daily level of nutrient intake that is sufficient to meet the requirements of nearly all (97-98%) healthy individuals in a particular life stage and gender group.
For children, calculating the RDA for vitamins like Vitamin A helps parents and caregivers understand how much is needed based on different factors, such as body weight. Utilizing models, such as the quadratic equation provided, is how researchers and scientists determine the necessary adjustments based on these specific factors.

Quadratic Formula

The quadratic formula is a mathematical tool used to solve quadratic equations, which typically have the form \(ax^2 + bx + c = 0\). A quadratic equation includes an \(x^2\) term and is known for its parabolic graphed shape.
The quadratic formula is expressed as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Here, \(a\), \(b\), and \(c\) are coefficients of the equation.

This formula works by calculating the solutions (or roots) of the equation. The expression under the square root sign \(b^2 - 4ac\) is called the discriminant. It determines the nature of the roots

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root.
• If negative, no real roots exist, and instead, there are two complex roots.

Using this formula, we derive solutions even when factoring is difficult or impossible.

Solving Quadratic Equations

Solving quadratic equations is a pivotal skill in algebra. These equations model a wide range of real-world scenarios, including the calculation of the RDA for vitamin A based on a child's weight.
To solve quadratic equations:

• First, ensure it is in standard form, \(ax^2 + bx + c = 0\).
• If factoring is feasible, use it to break down the equation into simpler binomials.
• When factoring isn't practical, the quadratic formula becomes essential.

Start by rewriting the equation if necessary, and ensure all terms are on one side of the equation. Apply the quadratic formula by substituting the values for \(a\), \(b\), and \(c\) to find \(x\). The calculated roots are the solutions to your equation, representing possible values for the variable \(x\), such as weight in pounds for the given scenario.
During practical problems like determining a child's necessary vitamin intake, interpreting these solutions necessitates choosing reasonable, context-appropriate results, especially when negative solutions do not fit within the realistic constraints of the problem.