Question

If \(y\) is directly proportional to \(x,\) and \(y\) has a value of 88.4 when \(x\) is 23.8 (a) Find the constant of proportionality. (b) Write the cquation \(y=f(x)\). (c) Find \(y\) when \(x=68.3\). (d) Find \(x\) when \(y=164\).

Short answer

The constant of proportionality (k) is 3.71428571429, the equation is y = 3.71428571429x, y is approximately 253.857142857 when x is 68.3, and x is approximately 44.1772151899 when y is 164.

Step-by-step solution

Understanding Direct Proportionality

When a quantity y is directly proportional to another quantity x, it means that as x increases, y increases by the same factor and vice versa. The relationship can be represented as y = kx, where k is the constant of proportionality.

Finding the Constant of Proportionality

To find the constant of proportionality (k), use the given values of y (88.4) and x (23.8) and set up the equation 88.4 = k * 23.8. Solve for k by dividing both sides by 23.8, giving k = 88.4 / 23.8.

Writing the Equation y=f(x)

Using the value of k found in Step 2, write the direct proportionality equation y = kx. Substitute k with its numeric value to get the specific function for y in terms of x.

Calculating y for x = 68.3

To find y when x is 68.3, substitute x = 68.3 into the equation y = kx and calculate y using the value of k found in Step 2.

Calculating x for y = 164

To find x when y is 164, substitute y = 164 into the equation y = kx and solve for x by dividing both sides by the value of k found in Step 2.

Key concepts

Constant of Proportionality

Understanding the concept of constant of proportionality is like deciphering the heartbeat of a direct relationship between two quantities. Imagine you have a recipe that for every one cup of sugar, calls for two cups of flour to maintain the perfect cake batter consistency. That fixed ratio between sugar and flour is akin to the constant of proportionality in mathematical terms.

When we say that one variable, let's call it 'y', is directly proportional to another, 'x', there's a special multiplier, which we denote as 'k', that always relates 'y' to 'x' in a neat and predictable way—much like our baking ratio. 'k' is what we call the constant of proportionality. In the textbook example, to find 'k' when given that 'y' equals 88.4 while 'x' is 23.8, the equation is essentially telling us how much 'y' increases for every single unit increase in 'x'. Through the simple division of 88.4 by 23.8, we unveil the constant of proportionality, and it becomes the backbone for making further predictions in our proportional relationship scenario.

In essence, 'k' is the key to unlocking the linear treasure that binds 'y' and 'x' together, making it considerably simpler to navigate through proportional puzzles.

Linear Relationship

A linear relationship is like a straight line drawn across a checked fabric, unwavering and constant in its slope, cutting across squares evenly. This steady line characterizes the relationship between two variables that are directly proportional. If you think about a ladder leaning against a wall, as you climb it rung by rung, your height from the ground increases at a constant rate; that's linear motion for you.

In mathematics, when two quantities have a linear relationship, changing one variable leads to a consistent change in another, much like the rungs of a ladder. For instance, if the cost of apples is directly proportional to their weight, doubling the weight would double the cost, and tripling the weight would triple the cost. There is a beautiful simplicity and predictability in this relationship.

The Graphical Perspective

Plotting such relationships on a graph yields a straight line, hence the name 'linear'. From the provided exercise, once the constant of proportionality is found, you can imagine plotting 'y' against 'x', picturing a straight path that tells an easy-to-follow story of how 'y' and 'x' dance together in harmony.

Proportional Relationship

In the world of mathematics, a proportional relationship is akin to a synchronized duet, with both partners moving in harmony, their steps matched to the tempo of the music. For two variables in a proportional relationship, their ratio to one another remains constant, much like how the rhythm of a song guides the consistent timing of dancers' movements.

Our textbook exercise showcases this harmonious relationship brilliantly. As the value of 'x' changes, 'y' follows suit in a way that their ratio, their dance steps, do not lose synchronicity. When one variable walks, the other runs alongside it, and neither gains nor loses ground on the other; they are in step. This proportional relationship is governed by the equation 'y = kx', where 'k' is our choreographer, the constant of proportionality, ensuring the ratio of 'y' to 'x' remains flawless.

In practical terms, this means that if you know the constant of proportionality and the value of one variable, the other variable can be readily calculated. Finding 'y' for a new 'x', or even solving for 'x' given a 'y', becomes a matter of plugging values into a simple equation and letting the math lead the dance.