What are the solutions to direct and inverse proportion questions including finding costs and proportionality constants?

$$\text{Cost of 12 pens} = \text{Cost per pen} \times 12 = \text{£}0.36 \times 12 = \boxed{\text{£}4.32}$$

2. 8 water bottles cost £20. Calculate the cost of 13 water bottles.

$$\text{Cost of 13 bottles} = \text{Cost per bottle} \times 13 = \text{£}2.50 \times 13 = \boxed{\text{£}32.50}$$

3. $X$ is directly proportional to the square of $Y$. When $X = 50$, $Y = 5$. Find $X$ when $Y = 3$.

$$X \propto Y^2$$

Introduce the constant of proportionality, $k$.

$$X = kY^2$$

$$50 = k(5^2)$$ $$50 = k(25)$$ $$k = \frac{50}{25}$$ $$k = 2$$

$$X = 2(3^2)$$ $$X = 2(9)$$ $$X = \boxed{18}$$

4. $C$ is directly proportional to the cube root of $D$. When $C = 32$, $D = 8$. Find $D$ when $C = 16$.

$$C \propto \sqrt[3]{D}$$

Introduce the constant of proportionality, $k$.

$$C = k\sqrt[3]{D}$$

$$32 = k\sqrt[3]{8}$$ $$32 = k(2)$$ $$k = \frac{32}{2}$$ $$k = 16$$

$$16 = 16\sqrt[3]{D}$$ $$\frac{16}{16} = \sqrt[3]{D}$$ $$1 = \sqrt[3]{D}$$

To find $D$, cube both sides.

$$D = 1^3$$ $$D = \boxed{1}$$

5. $X$ is inversely proportional to the square of $Y$. When $X = 2$, $Y = 5$. Find $X$ when $Y = \sqrt{10}$.

$$X \propto \frac{1}{Y^2}$$

Introduce the constant of proportionality, $k$.

$$X = \frac{k}{Y^2}$$

$$2 = \frac{k}{5^2}$$ $$2 = \frac{k}{25}$$ $$k = 2 \times 25$$ $$k = 50$$

$$X = \frac{50}{(\sqrt{10})^2}$$ $$X = \frac{50}{10}$$ $$X = \boxed{5}$$