Algebra - Proportion

$y$ is directly proportional to $x$. Given that $y = 15$ when $x = 3$, find the value of $y$ when $x = 10$.

Step 1: Write the general equation. $y = kx$ Step 2: Substitute the known values to find $k$. $15 = k \times 3$ $k = \frac{15}{3} = 5$ Step 3: Write the specific formula. $y = 5x$ Step 4: Substitute $x = 10$ to find $y$. $y = 5 \times 10 = 50$

Since $y$ is directly proportional to $x$, we use the linear relationship $y=kx$. We find the constant $k$ first and then use it to find the unknown value.

$h$ is inversely proportional to $r^2$. When $r = 2$, $h = 9$. Find $h$ when $r = 3$.

Step 1: Write the general equation. $h = \frac{k}{r^2}$ Step 2: Substitute the known values to find $k$. $9 = \frac{k}{2^2}$ $9 = \frac{k}{4}$ $k = 36$ Step 3: Write the specific formula. $h = \frac{36}{r^2}$ Step 4: Substitute $r = 3$ to find $h$. $h = \frac{36}{3^2}$ $h = \frac{36}{9} = 4$

Inverse proportion means $h$ is equal to $k$ divided by the square of $r$. Once the constant $k=36$ is established, we solve for $h$ using the new value of $r$.

The time $T$ taken for a pendulum to swing is proportional to the square root of its length $L$. If $T = 2$ seconds when $L = 16$ cm, find $T$ when $L = 64$ cm.

Step 1: Write the equation. $T = k\sqrt{L}$ Step 2: Find $k$. $2 = k\sqrt{16}$ $2 = k \times 4$ $k = \frac{2}{4} = 0.5$ Step 3: Substitute $L = 64$. $T = 0.5 \times \sqrt{64}$ $T = 0.5 \times 8 = 4$

This is a direct proportion involving a square root. We solve for $k$ using the square root of the initial length, then apply it to the new length.