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Chapter 7: Problem 75

Find an equation of variation in which: \(y\) varies jointly as \(w\) and the square of \(x\) and inversely as \(z,\) and \(y=49\) when \(w=3, x=7\) and \(z=12\)

Short Answer

Expert verified
The equation is \( y = 4 \frac{wx^2}{z} \).

Step by step solution

01

Write the General Form of the Variation Equation

To write the equation that describes how the variable varies, combine all given conditions. Since \(y\) varies jointly as \(w\) and the square of \(x\) and inversely as \(z\), we can write: \[ y = k \frac{wx^2}{z} \] where \(k\) is the constant of proportionality.
02

Substitute Given Values to Find the Constant of Proportionality

Using the given values \(y=49, w=3, x=7, z=12\), substitute into the general form of the equation to find \(k\): \[ 49 = k \frac{3 \cdot 7^2}{12} \]
03

Simplify the Equation to Solve for k

First, simplify inside the fraction: \[ 7^2 = 49 \] and then: \[ 3 \cdot 49 = 147 \] Therefore: \[ 49 = k \frac{147}{12} \] To isolate \(k\), multiply both sides by \(\frac{12}{147}\): \[ k = 49 \cdot \frac{12}{147} \]
04

Perform the Multiplication and Simplification

Calculate the value of \(k\): \[ 49 \cdot \frac{12}{147} = 4 \] Thus, we have: \[ k = 4 \]
05

Write the Final Equation

Now, substitute the value of \(k\) back into the general form of the equation: \[ y = 4 \frac{wx^2}{z} \]

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Key Concepts

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

proportionality constant
In the context of variation problems, the proportionality constant, often represented as 'k', is a crucial element. It links the relationship between variables. For instance, in the given exercise, we determined that the constant of proportionality is 4 after substituting the provided values into the initial variation equation and solving for 'k'. This constant essentially scales the proportional relationship. Without it, the joint and inverse relationships cannot be accurately described. Remember, finding 'k' requires known values of all other variables in the equation.
variation equation
A variation equation expresses how one variable changes in relation to others. In our specific problem, we used the formula \( y = k \frac{wx^2}{z} \) to describe how 'y' varies with 'w', 'x', and 'z'. These equations can represent direct, inverse, or joint variation.
The structure of a variation equation consists of:
  • The dependent variable ('y' in this case).
  • The proportionality constant ('k').
  • The independent variables, which may contribute through multiplication (indicating direct or joint variation) or division (indicating inverse variation).
This form is essential for understanding and solving real-world problems involving dependent relationships.
inverse variation
Inverse variation occurs when one variable increases while another decreases proportionally. In mathematical terms, if 'y' is inversely proportional to 'z', then the product of 'y' and 'z' is constant (\( y \times z = k \)). In the exercise, 'y' varies inversely as 'z', implying that as 'z' gets larger, 'y' gets smaller if all else remains constant. Understanding inverse variation helps solve problems where increases in one variable result in decreases in another. Always check units and proportional relationships to accurately apply this concept.
joint variation
'Joint variation' describes a scenario where a variable depends on two or more other variables together. In our exercise, 'y' varies jointly as 'w' and the square of 'x' (\( x^2 \)) and inversely as 'z'. This can be written in a combined equation: \( y = k \frac{wx^2}{z} \). Joint variation implies that 'y' will increase if either 'w' or \( x^2 \) increases, but will decrease if 'z' increases. Real-world examples include physical laws like those governing force, pressure, or gravitational attraction, where multiple factors influence the outcome together.

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Most popular questions from this chapter

The volume \(V\) of a given mass of a gas varies directly as the temperature \(T\) and inversely as the pressure \(P .\) If \(V=231 \mathrm{cm}^{3}\) when \(T=300^{\circ} \mathrm{K}\) (Kelvin) and \(P=20 \mathrm{lb} / \mathrm{cm}^{2},\) what is the volume when \(T=320^{\circ} \mathrm{K}\) and \(P=16 \mathrm{Ib} / \mathrm{cm}^{2} ?\)

Escalators. Together, a \(100-\mathrm{cm}\) wide escalator and a \(60-\mathrm{cm}\) wide escalator can empty a 1575 -person auditorium in 14 min. The wider escalator moves twice as many people as the narrower one. How many people per hour does the \(60-\mathrm{cm}\) wide escalator move?

One factor influencing urban planning is VMT, or vehicle miles traveled. The table below lists the annual VMT per household for various densities for a typical urban area. $$\begin{array}{c|c}{\text { Population Density }} \\ {\text { (in number of households }} & {\text { Annual VMT }} \\ { \text { per residential acre } )} & {\text { per Household }} \\ {25} & {12,000} \\ {50} & {6,000} \\ {100} & {3,000} \\ {200} & {1,500}\end{array}$$ a) Determine whether the data indicate direct variation or inverse variation. b) Find an equation of variation that describes the data. c) Use the equation to estimate the annual VMT per household for areas with 10 households per residential acre.

The product of two consecutive even integers is 48. Find the numbers.

Paloma drove \(200 \mathrm{km} .\) For the first \(100 \mathrm{km}\) of the trip, she drove at a speed of \(40 \mathrm{km} / \mathrm{h}\). For the second half of the trip, she traveled at a speed of \(60 \mathrm{km} / \mathrm{h}\). What was the average speed of the entire trip? (It was \(n o t\) \(50 \mathrm{km} / \mathrm{h} .)\)
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