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Chapter 18: Problem 21

Find the required value by setting up the general equation and then evaluating. Find \(y\) when \(x=10\) if \(y\) varies directly as \(x,\) and \(y=200\) when \(x=80\)

Short Answer

Expert verified
The value of \( y \) when \( x = 10 \) is 25.

Step by step solution

01

Identifying Direct Variation

In problems where one variable varies directly as another, we use the relationship: \( y = kx \), where \( k \) is the constant of proportionality. This means \( y \) changes in proportion to \( x \).
02

Finding the Constant of Proportionality

We know \( y = 200 \) when \( x = 80 \). Using the direct variation equation, substitute these values to find \( k \): \( 200 = k \times 80 \). Solve for \( k \) by dividing both sides by 80: \( k = \frac{200}{80} = 2.5 \).
03

Setting Up the Equation

Using the value of \( k \) found, substitute it back into the direct variation equation: \( y = 2.5x \).
04

Substituting x = 10

We need to find \( y \) when \( x = 10 \). Substitute \( x = 10 \) into the equation \( y = 2.5x \): \( y = 2.5 \times 10 \).
05

Calculating y for x = 10

Complete the multiplication to find \( y \): \( y = 25 \).

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Key Concepts

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

Constant of Proportionality
When we talk about direct variation, the concept of the 'constant of proportionality' is key. It is represented by the symbol \( k \) in the equation \( y = kx \). Here, \( y \) and \( x \) are two variables that have a special relationship. They change together in a predictable way.
  • If \( x \) increases, \( y \) increases too.
  • If \( x \) decreases, so does \( y \).
The constant of proportionality tells us exactly how much \( y \) changes when \( x \) changes.
In our exercise, we found \( k \) by using known values for \( y \) and \( x \). When \( y = 200 \) and \( x = 80 \), we calculated \( k \) as \( \frac{200}{80} = 2.5 \). This number tells us that for every unit increase in \( x \), \( y \) increases by 2.5 units.
Proportional Relationship
A proportional relationship is a special type of relationship between two variables. It happens when two things change at a constant rate. In math, this is often expressed as \( y = kx \).
With a proportional relationship, as one thing gets bigger, the other one does too, at a rate that's predictable. The beauty of this relationship is its simplicity.
  • It means if you know one value, you can always find the other.
  • The graph of this relationship is a straight line through the origin.
In the exercise, the relationship was given as direct, meaning every change in \( x \) had a consistent effect on \( y \). This straightforwardness makes proportional relationships easy to work with. You only need \( k \) and one pair of \( x \) and \( y \) to understand the whole relationship.
Mathematical Equations
Mathematical equations are the tools that translate real-life situations into mathematical terms. In our problem, the equation \( y = kx \) models the direct variation between \( y \) and \( x \).
Equations use symbols and numbers to express relationships and patterns.
  • They help solve problems by setting up precise relationships.
  • They allow us to predict unknown values.
In the problem, the steps outlined guide us from identifying the type of variation, to finding \( k \), and using it to predict unknown values. Once the equation \( y = 2.5x \) was established, finding \( y \) for \( x = 10 \) became straightforward.
Mathematical equations like these help visualize and solve direct variation problems with ease.

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Most popular questions from this chapter

Solve the given applied problems involving variation. The power gain \(G\) by a parabolic microwave dish varies directly as the square of the diameter \(d\) of the opening and inversely as the square of the wavelength \(\lambda\) of the wave carrier. Find the equation relating \(G, d,\) and \(\lambda\) if \(G=5.5 \times 10^{4}\) for \(d=2.9 \mathrm{m}\) and \(\lambda=3.0 \mathrm{cm}\)

Give the specific equation relating the variables after evaluating the constant of proportionality for the given set of values. \(v\) is proportional to \(t\) and the square root of \(s,\) and \(v=80\) when \(s=4\) and \(t=5\)

$$\text { Find the required ratios.}$$ The electric current in a given circuit is the ratio of the voltage to the resistance. What is the current \((1 \mathrm{V} / 1 \Omega=1 \mathrm{A})\) for a circuit where the voltage is \(24.0 \mathrm{mV}\) and the resistance is \(10.0 \Omega ?\)

Answer the given questions by setting up and solving the appropriate proportions. A motorist traveled \(5.0 \mathrm{km}\) (accurately measured) in \(4 \mathrm{min} 54 \mathrm{s}\) and the speedometer showed \(58 \mathrm{km} / \mathrm{h}\) for the same interval. What is the percent error in the speedometer reading?

Find the required ratios. An important design feature of an aircraft wing is its aspect ratio. It is defined as the ratio of the square of the span of the wing (wingtip to wingtip) to the total area of the wing. If the span of the wing for a certain aircraft is \(32.0 \mathrm{ft}\) and the area is \(195 \mathrm{ft}^{2}\), find the aspect ratio.
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