Problem 47 - Precalculus with Limits

Question

The coiled spring of a toy supports the weight of a child. The spring is compressed a distance of 1.9 inches by the weight of a 25-pound child. The toy will not work properly if its spring is compressed more than 3 inches. What is the weight of the heaviest child who should be allowed to use the toy?

Step-by-Step Solution

Answer

The heaviest child who should be allowed to use the toy weighs approximately 39.47 pounds.

Step 1: Identify and list down given values

From the exercise, we know that a 25-pound child compresses the spring 1.9 inches, and that the spring can be compressed a maximum of 3 inches.

Step 2: Formulate the equation

We can formulate an equation using these values, creating a proportion, where weight over compression in one situation equals weight over compression in another. In this case, the proportion is \(\frac{25}{1.9} = \frac{x}{3}\), where x is the unknown weight.

Step 3: Solve for x

Now, perform cross multiplication and solve for x. You should get \(x = \frac{25*3}{1.9}\).

Step 4: Evaluate and round off if necessary

Evaluate the expression, ensuring accuracy up to decimal places. Depending on the requirement, you may need to round off the final answer.

Key Concepts

Spring Compression

Spring compression is a fundamental concept in physics relating to mechanics and elasticity. When a spring is compressed, energy is stored within the spring as potential energy. Hooke's Law, which states that the force needed to compress or extend a spring by some distance is proportional to that distance. This is depicted by the equation \( F = k \times x \), where \( F \) is the force applied to the spring, \( k \) is the spring constant (a measure of the spring's stiffness), and \( x \) is the displacement from the spring's original length.

In practical applications, such as the toy spring in the exercise mentioned, the amount of weight (force) applied to the spring results in a corresponding compression. Understanding the relationship between force and spring compression is crucial when designing mechanical systems which include springs, to ensure that they function safely and efficiently. For the toy described, it is indicated that if a spring is compressed more than 3 inches, the toy would not operate correctly. Therefore, understanding the limit of spring compression is vital for determining the maximum weight it can support safely.

Proportional Reasoning

Understanding Ratios and Proportions

Proportional reasoning is the process of understanding the relationship between quantities that change in a way that is related to one another by a constant multiplier. It involves the use of ratios and proportions to solve problems and make predictions.

Considering the toy example, if a child weighing 25 pounds causes the spring to compress by 1.9 inches, the ratio of weight to compression (25/1.9) remains constant, even for different weights and corresponding compressions. This constant ratio helps us determine the maximum weight that does not over-compress the spring, by setting up a proportion based on the understood maximum compression of 3 inches. Proportional reasoning allows one to predict that a heavier child will compress the spring more.

Importance in Scientific and Everyday Contexts

Aside from mechanical applications, proportional reasoning is vital in various fields like chemistry, economics, and even daily tasks such as cooking or budgeting. The ability to comprehend and apply proportional relationships is a key mathematical skill.

Algebraic Equation

Formulating Equations

An algebraic equation is a statement of equality that involves variables. It represents a problem that requires a solution, or an unknown quantity which can be found through algebraic manipulation. In the case of the spring compression scenario, this manipulation involves creating an equation where the weights of the children are directly proportional to the compression of the spring.

To find the unknown weight, we create the equation \( \frac{25}{1.9} = \frac{x}{3} \), where 25 pounds is the known weight, 1.9 inches is the known compression, 3 inches is the maximum allowed compression, and \( x \) represents the unknown maximum weight. The solution to this equation yields the weight of the heaviest child that can safely use the toy without over-compressing the spring.

Role in Problem Solving

The use of algebraic equations is at the heart of solving many scientific, engineering, and mathematical problems, as it provides a structured method to figure out unknowns in a given system or scenario.

Cross Multiplication

Solving Proportions

Cross multiplication is a method for solving proportions where two ratios are equal. By multiplying the cross-products of the numerators and denominators, you can solve for the unknown variable. Specifically, the product of the means equals the product of the extremes. In this instance, we have the proportion \( \frac{25}{1.9} = \frac{x}{3} \), where cross multiplying gives us \( 25 \times 3 = 1.9 \times x \).

Practical Application

Once cross multiplied, we can solve for \( x \) by dividing both sides of the equation by 1.9, resulting in \( x = \frac{25 \times 3}{1.9} \). This step is fundamental as it simplifies the process of finding 'x' without the need for complicated algebraic manipulation. Cross multiplication is not only a powerful technique for solving textbook problems but is also widely used in various practical situations, such as scaling recipes, creating models, or adjusting proportions to fit different circumstances.