DV and IV Combined!

A. Using k as the constant of variation, write the equation of variation for each of the following

1. W varies jointly as c and the square of a and inversely as b.

2. P varies directly as the square of x and inversely as s.

3. The electrical resistance R of a wire varies directly as its length I and inversely as the square of its diameter d.

4. The acceleration A of a moving object varies directly as the distance d it travels and inversely as the square of the time r it travels.

5. The pressure P of a gas varies directly as its temperature 1 and inversely as its volume V.​

In each of the following cases, the variable in question varies directly or inversely as specified. We will write the corresponding variation equation using \( k \) as the constant of variation.

### 1. \( W \) varies jointly as \( c \) and the square of \( a \) and inversely as \( b \).
\[
W = k \frac{c \cdot a^2}{b}
\]

### 2. \( P \) varies directly as the square of \( x \) and inversely as \( s \).
\[
P = k \frac{x^2}{s}
\]

### 3. The electrical resistance \( R \) of a wire varies directly as its length \( l \) and inversely as the square of its diameter \( d \).
\[
R = k \frac{l}{d^2}
\]

### 4. The acceleration \( A \) of a moving object varies directly as the distance \( d \) it travels and inversely as the square of the time \( r \) it travels.
\[
A = k \frac{d}{r^2}
\]

### 5. The pressure \( P \) of a gas varies directly as its temperature \( T \) and inversely as its volume \( V \).
\[
P = k \frac{T}{V}
\]

Let me know if you need further clarification or details!

### Related Questions:
1. What is the significance of the constant \( k \) in each variation equation?
2. How does a joint variation differ from a direct or inverse variation?
3. Can you derive the formula for a situation where \( W \) varies inversely as both \( b \) and \( d^2 \)?
4. How does changing the temperature affect the pressure in the gas variation equation?
5. In the second equation, if \( s \) doubles, how does \( P \) change?

### Tip:
In joint variations, remember that variables can influence the outcome both directly and inversely. Organizing them correctly ensures you write the equation accurately.

DV and IV Combined!
A. Using k as the constant of variation, write the equation of variation for each of the following

1. W varies jointly as c and the square of a and inversely as b.
2. P varies directly as the square of x and inversely as s.
3. The electrical resistance R of a wire varies directly as its length l and inversely as the square of its diameter d.
4. The acceleration A of a moving object varies directly as the distance d it travels and inversely as the square of the time r it travels.
5. The pressure P of a gas varies directly as its temperature T and inversely as its volume V.

Explore how to solve problems involving direct, inverse, and joint variations. Learn how variables interact in equations, such as W varies jointly as c and a^2 and inversely as b, with real-life physics applications. Suitable for Grades 9-12.

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