The provided image shows a right-angled triangular prism. We are tasked with calculating:

1. The volume of the prism.
2. The surface area of the prism.

Given Data:

• Right triangle ABC forms the base of the prism.
• $AB = 3 \, \text{cm}$ (one leg of the right triangle).
• $\angle ABC = 42^\circ$.
• The prism has a length (distance between the two triangular faces) of $10 \, \text{cm}$.

Steps to Solve:

1. Volume of the Prism:

The volume of a prism is given by the formula: $\text{Volume} = \text{Base Area} \times \text{Length}$

For this triangular prism, we first need to calculate the area of the triangular base $\triangle ABC$.

• To find the height $BC$ (opposite the $42^\circ$ angle), we use the trigonometric relation: $\sin(42^\circ) = \frac{BC}{AB}$ Thus, $BC = AB \times \sin(42^\circ) = 3 \times \sin(42^\circ)$
• Similarly, to find the adjacent side $AC$, use: $AC = AB \times \cos(42^\circ) = 3 \times \cos(42^\circ)$

Once we have both $BC$ and $AC$, the area of triangle $ABC$ is: $\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times BC$

Finally, multiplying this base area by the prism's length (10 cm) gives the volume.

2. Surface Area of the Prism:

The surface area of the prism consists of:

• Two triangular faces $ABC$ and $EDC$.
• Three rectangular faces: $ABED$, $BCDE$, and $ACDE$.

We calculate the areas of these individual faces:

• The area of the triangular faces $ABC$ is already calculated.
• The areas of the three rectangular faces are calculated using the sides of the triangle and the prism's length:
  • $ABED$ has dimensions $AB \times 10$.
  • $BCDE$ has dimensions $BC \times 10$.
  • $ACDE$ has dimensions $AC \times 10$.

Adding all the areas gives the total surface area of the prism.

Let me calculate these values for you.### Solutions:

1. Volume of the prism: $30.11 \, \text{cm}^3$
2. Surface area of the prism: $78.39 \, \text{cm}^2$

Would you like further clarification on any steps or additional details?

Here are 5 related questions for further practice:

1. How would the volume change if the length of the prism were doubled?
2. Can you calculate the height of the triangle $BC$ in terms of the prism's dimensions?
3. How would the surface area change if the angle $\angle ABC$ were increased?
4. What would the area of one of the rectangular faces be if the length of the prism were increased by 5 cm?
5. How would you calculate the volume of the prism if the base were not a right triangle?

Tip: When calculating the surface area of a prism, always ensure to account for all faces, including the base and lateral faces.