Number - Constant of proportionality and graphs of direct and inverse proportion-extended

The variable $y$ is directly proportional to the square of $x$. When $x = 3$, $y = 18$. Find the value of $y$ when $x = 5$.

1. Set up the equation: $y = kx^2$
2. Substitute $x = 3$ and $y = 18$: $18 = k(3^2)$ $18 = 9k$ $k = \frac{18}{9} = 2$
3. Write the specific equation: $y = 2x^2$
4. Substitute $x = 5$: $y = 2(5^2)$ $y = 2(25) = 50$

Since $y$ is proportional to $x^2$, we find the constant $k$ first using the given values, then use that constant to calculate the new value of $y$.

The pressure $P$ of a gas is inversely proportional to its volume $V$. When $V = 4$ $m^3$, $P = 100$ $Pa$. Find the pressure when the volume is $8$ $m^3$.

1. Set up the equation: $P = \frac{k}{V}$
2. Substitute $V = 4$ and $P = 100$: $100 = \frac{k}{4}$ $k = 100 \times 4 = 400$
3. Write the specific equation: $P = \frac{400}{V}$
4. Substitute $V = 8$: $P = \frac{400}{8} = 50$

In inverse proportion, the product of the variables is constant ($P \times V = k$). Increasing the volume results in a proportional decrease in pressure.

A graph of $y$ against $x$ is a straight line through the origin. If the line passes through the point $(4, 12)$, determine the constant of proportionality and the equation of the line.

1. Since the graph is a straight line through the origin, $y = kx$.
2. Substitute the coordinates $(4, 12)$: $12 = k(4)$ $k = \frac{12}{4} = 3$
3. The equation is $y = 3x$.

The gradient of a direct proportion graph is the constant of proportionality $k$. Use the point $(x, y)$ to find the slope.

1. Set up the equation: $y = kx^3$.
2. Substitute the point $(2, 40)$: $40 = k(2)^3$.
3. Solve for $k$: $40 = 8k \implies k = 5$.
4. Find $y$ when $x = 3$: $y = 5(3)^3 = 5 \times 27 = 135$.

Since $y$ is proportional to $x^3$, we use the cubic model. Substituting the known coordinates allows us to find the constant $k$, which remains the same for all points on this curve.

1. State the relationship: $I = \frac{k}{d^2}$.
2. Find $k$: $20 = \frac{k}{3^2} \implies 20 = \frac{k}{9} \implies k = 180$.
3. Formulate the equation: $I = \frac{180}{d^2}$.
4. Calculate for $d = 5$: $I = \frac{180}{5^2} = \frac{180}{25} = 7.2$ units.

This follows the inverse square law. We first determine the constant of proportionality by substituting the initial conditions, then use the constant to evaluate the second state.