Let's solve the problem in the image step by step.

Part (a)

The equation is:

$\sqrt[3]{\frac{p}{q}} + 2 = 4$

To express $p$ in terms of $q$, let's follow these steps:

1. Isolate the cube root term: $\sqrt[3]{\frac{p}{q}} = 4 - 2$ $\sqrt[3]{\frac{p}{q}} = 2$

2. Cube both sides to eliminate the cube root: $\frac{p}{q} = 2^3$ $\frac{p}{q} = 8$

3. Solve for $p$: $p = 8q$

Thus, $p$ in terms of $q$ is:

$p = 8q$

Part (b)

The formula for the volume $V$ of the prism is given by:

$V = \frac{w \times h \times L}{2}$

(i) What does this formula represent?

This formula represents the volume of a triangular prism, where:

• $w$ is the width of the triangular base,
• $h$ is the height of the triangular base,
• $L$ is the length of the prism.

The area of the triangular base is $\frac{w \times h}{2}$, and the volume of the prism is the area of the base multiplied by its length $L$.

(ii) Given $V = 300 \, \text{cm}^3$, $w = 4 \, \text{cm}$, and $L = 30 \, \text{cm}$, find $h$.

We are given:

• $V = 300 \, \text{cm}^3$,
• $w = 4 \, \text{cm}$,
• $L = 30 \, \text{cm}$.

Using the volume formula:

$300 = \frac{4 \times h \times 30}{2}$

Simplify:

$300 = \frac{120h}{2}$ $300 = 60h$

Solve for $h$:

$h = \frac{300}{60} = 5 \, \text{cm}$

Thus, the height $h$ is $5 \, \text{cm}$.

Summary of Answers:

• Part (a): $p = 8q$
• Part (b):
  • (i) The formula represents the volume of a triangular prism.
  • (ii) $h = 5 \, \text{cm}$.

Would you like any further explanation on these steps?

Follow-up Questions:

1. How do you find the volume of other types of prisms?
2. What happens if $p$ is negative in part (a)?
3. Can you explain why we cube both sides in part (a)?
4. How would the formula change if the base of the prism were a rectangle?
5. What are some other real-life examples where triangular prisms are used?

Tip:

When solving equations with cube roots or square roots, always isolate the root term first before performing the inverse operation (cubing or squaring).